Lectures in logic and set theory. Mathematical logic

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS EDITORIAL BOARD B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK Lectures in Logic and Set Theory Volume 1 This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume 1 includes formal proof techniques, a section on applications of compactness (including non-standard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of G¨odel’s second incompleteness theorem since Hilbert and Bernay’s Grundlagen.

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K. Petersen Ergodic theory P.T. Johnstone Stone spaces J.-P. Kahane Some random series of functions, 2nd edition J. Lambek & P.J. Scott Introduction to higher-order categorical logic H. Matsumura Commutative ring theory M. Aschbacher Finite group theory, 2nd edition J.L. Alperin Local representation theory P. Koosis The logarithmic integral I S.J. Patterson An introduction to the theory of the Riemann zeta-function H.J. Baues Algebraic homotopy V.S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups W. Dicks & M. Dunwoody Groups acting on graphs R. Fritsch & R. Piccinini Cellular structures in topology H. Klingen Introductory lectures on Siegel modular forms P. Koosis The logarithmic integral II M.J. Collins Representations and characters of finite groups H. Kunita Stochastic flows and stochastic differential equations P. Wojtaszczyk Banach spaces for analysts J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis A. Frohlich & M.J. Taylor Algebraic number theory K. Goebel & W.A. Kirk Topics in metric fixed point theory J.F. Humphreys Reflection groups and Coxeter groups D.J. Benson Representations and cohomology I D.J. Benson Representations and cohomology II C. Allday & V. Puppe Cohomological methods in transformation groups C. Soule et al. Lectures on Arakelov geometry A. Ambrosetti & G. Prodi A primer of nonlinear analysis J. Palis & F. Takens Hyperbolicity, stability and chaos at homoclinic bifurcations Y. Meyer Wavelets and operators 1 C. Weibel, An introduction to homological algebra W. Bruns & J. Herzog Cohen-Macaulay rings V. Snaith Explicit Brauer induction G. Laumon Cohomology of Drinfeld modular varieties I E.B. Davies Spectral theory and differential operators J. Diestel, H. Jarchow, & A. Tonge Absolutely summing operators P. Mattila Geometry of sets and measures in Euclidean spaces R. Pinsky Positive harmonic functions and diffusion G. Tenenbaum Introduction to analytic and probabilistic number theory C. Peskine An algebraic introduction to complex projective geometry Y. Meyer & R. Coifman Wavelets R. Stanley Enumerative combinatorics I I. Porteous Clifford algebras and the classical groups M. Audin Spinning tops V. Jurdjevic Geometric control theory H. Volklein Groups as Galois groups J. Le Potier Lectures on vector bundles D. Bump Automorphic forms and representations G. Laumon Cohomology of Drinfeld modular varieties II D.M. Clark & B.A. Davey Natural dualities for the working algebraist J. McCleary A user’s guide to spectral sequences II P. Taylor Practical foundations of mathematics M.P. Brodmann & R.Y. Sharp Local cohomology J.D. Dixon et al. Analytic pro-P groups R. Stanley Enumerative combinatorics II R.M. Dudley Uniform central limit theorems J. Jost & X. Li-Jost Calculus of variations A.J. Berrick & M.E. Keating An introduction to rings and modules S. Morosawa Holomorphic dynamics A.J. Berrick & M.E. Keating Categories and modules with K-theory in view K. Sato Levy processes and infinitely divisible distributions H. Hida Modular forms and Galois cohomology R. Iorio & V. Iorio Fourier analysis and partial differential equations R. Blei Analysis in integer and fractional dimensions F. Borceaux & G. Janelidze Galois theories B. Bollobas Random graphs

LECTURES IN LOGIC AND SET THEORY Volume 1: Mathematical Logic

GEORGE TOURLAKIS York University

   Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521753739 © George Tourlakis 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 ISBN-13 978-0-511-06871-3 eBook (EBL) ISBN-10 0-511-06871-9 eBook (EBL) ISBN-13 978-0-521-75373-9 hardback ISBN-10 0-521-75373-2 hardback

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για την δεσπoινα, την µαρινα και τoν γιαννη

Contents

Preface Basic Logic I.1 First Order Languages I.2 A Digression into the Metatheory: Informal Induction and Recursion I.3 Axioms and Rules of Inference I.4 Basic Metatheorems I.5 Semantics; Soundness, Completeness, Compactness I.6 Substructures, Diagrams, and Applications I.7 Defined Symbols I.8 Computability and Uncomputability I.9 Arithmetic, Definability, Undefinability, and Incompletableness I.10 Exercises II The Second Incompleteness Theorem II.1 Peano Arithmetic II.2 A Formal β-Function II.3 Formal Primitive Recursion II.4 The Boldface  and  II.5 Arithmetization II.6 Derivability Conditions; Fixed Points II.7 Exercises Bibliography List of Symbols Index I

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page ix 1 5 19 28 42 52 75 112 123 155 191 205 206 232 248 256 265 272 316 319 321 323

Preface

Both volumes in this series are about what mathematicians, especially logicians, call the “foundations” (of mathematics) – that is, the tools of the axiomatic method, an assessment of their effectiveness, and two major examples of application of these tools, namely, in the development of number theory and set theory. There have been, in hindsight, two main reasons for writing this volume. One was the existence of notes I wrote for my lectures in mathematical logic and computability that had been accumulating over the span of several years and badly needed sorting out. The other was the need to write a small section on logic, “A Bit of Logic” as I originally called it, that would bootstrap my volume on set theory† on which I had been labouring for a while. Well, one thing led to another, and a 30 or so page section that I initially wrote for the latter purpose grew to become a self-standing volume of some 300 pages. You see, this material on logic is a good story and, as with all good stories, one does get carried away wanting to tell more. I decided to include what many people will consider, I should hope, as being the absolutely essential topics in proof, model, and recursion theory – “absolutely essential” in the context of courses taught near the upper end of undergraduate, and at the lower end of graduate curricula in mathematics, computer science, or philosophy. But no more.‡ This is the substance of Chapter I; hence its title “Basic Logic”. † ‡

A chapter by that name now carries out these bootstrapping duties – the proverbial “Chapter 0” (actually Chapter I) of volume 2. These topics include the foundation and development of non-standard analysis up to the extreme value theorem, elementary equivalence, diagrams, and L¨owenheim-Skolem theorems, and G¨odel’s first incompleteness theorem (along with Rosser’s sharpening).

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But then it occurred to me to also say something about one of the most remarkable theorems of logic – arguably the most remarkable – about the limitations of formalized theories: G¨odel’s second incompleteness theorem. Now, like most reasonable people, I never doubted that this theorem is true, but, as the devil is in the details, I decided to learn its proof – right from Peano’s axioms. What better way to do this than writing down the proof, gory details and all? This is what Chapter II is about.† As a side effect, the chapter includes many theorems and techniques of one of the two most important – from the point of view of foundations – “applied” logics (formalized theories), namely, Peano arithmetic (the other one, set theory, taking all of volume 2). I have hinted above that this (and the second) volume are aimed at a fairly advanced reader: The level of exposition is designed to fit a spectrum of mathematical sophistication from third year undergraduate to junior graduate level (each group will find here its favourite sections that serve its interests and level of preparation – and should not hesitate to judiciously omit topics). There are no specific prerequisites beyond some immersion in the “proof culture”, as this is attainable through junior level courses in calculus, linear algebra, or discrete mathematics. However, some familiarity with concepts from elementary na¨ıve set theory such as finiteness, infinity, countability, and uncountability will be an asset.‡ A word on approach. I have tried to make these lectures user-friendly, and thus accessible to readers who do not have the benefit of an instructor’s guidance. Devices to that end include anticipation of questions, frequent promptings for the reader to rethink an issue that might be misunderstood if glossed over (“Pauses”), and the marking of important passages, by , as well as those that . can be skipped at first reading, by Moreover, I give (mostly) very detailed proofs, as I know from experience that omitting details normally annoys students. †

‡

It is strongly conjectured here that this is the only complete proof in print other than the one that was given in Hilbert and Bernays (1968). It is fair to clarify that I use the term “complete proof” with a strong assumption in mind: That the axiom system we start with is just Peano arithmetic. Proofs based on a stronger – thus technically more convenient – system, namely, primitive recursive arithmetic, have already appeared in print (Diller (1976), Smory´nski (1985)). The difficulty with using Peano arithmetic as the starting point is that the only primitive recursive functions initially available are the successor, identity, plus, and times. An awful amount of work is needed – a preliminary “coding trick” – to prove that all the rest of the primitive recursive functions “exist”. By then are we already midway in Chapter II, and only then are we ready to build G¨odel numbers of terms, formulas, and proofs and to prove the theorem. I have included a short paragraph nicknamed “a crash course on countable sets” (Section I.5, p. 62), which certainly helps. But having seen these topics before helps even more.

Preface

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The first chapter has a lot of exercises (the second having proportionally fewer). Many of these have hints, but none are marked as “hard” vs. “just about right”, a subjective distinction I prefer to avoid. In this connection here is some good advice I received when I was a graduate student at the University of Toronto: “Attempt all the problems. Those you can do, don’t do. Do the ones you cannot”. What to read. Consistently with the advice above, I suggest that you read this volume from cover to cover – including footnotes! – skipping only what you already know. Now, in a class environment this advice may be impossible to take, due to scope and time constraints. An undergraduate (one semester) course in logic at the third year level will probably cover Sections I.1–I.5, making light of Section I.2, and will introduce the student to the elements of computability along with a hand-waving “proof” of G¨odel’s first incompleteness theorem (the “semantic version” ought to suffice). A fourth year class will probably attempt to cover the entire Chapter I. A first year graduate class has no more time than the others at its disposal, but it usually goes much faster, skipping over familiar ground, thus it will probably additionally cover Peano arithmetic and will get to see how G¨odel’s second theorem follows from L¨ob’s derivability conditions. Acknowledgments. I wish to offer my gratitude to all those who taught me, a group led by my parents and too large to enumerate. I certainly include my students here. I also include Raymond Wilder’s book on the foundations of mathematics, which introduced me, long long ago, to this very exciting field and whetted my appetite for more (Wilder (1963)). I should like to thank the staff at Cambridge University Press for their professionalism, support, and cooperation, with special appreciation due to Lauren Cowles and Caitlin Doggart, who made all the steps of this process, from refereeing to production, totally painless. This volume is the last installment of a long project that would have not been successful without the support and warmth of an understanding family (thank you). I finally wish to record my appreciation to Donald Knuth and Leslie Lamport for the typesetting tools TEX and LATEX that they have made available to the technical writing community, making the writing of books such as this one almost easy. George Tourlakis Toronto, March 2002

I Basic Logic

Logic is the science of reasoning. Mathematical logic applies to mathematical reasoning – the art and science of writing down deductions. This volume is about the form, meaning, use, and limitations of logical deductions, also called proofs. While the user of mathematical logic will practise the various proof techniques with a view of applying them in everyday mathematical practice, the student of the subject will also want to know about the power and limitations of the deductive apparatus. We will find that there are some inherent limitations in the quest to discover truth by purely formal – that is, syntactic – techniques. In the process we will also discover a close affinity between formal proofs and computations that persists all the way up to and including issues of limitations: Not only is there a remarkable similarity between the types of respective limitations (computations vs. uncomputable functions, and proofs vs. unprovable, but “true”, sentences), but, in a way, you cannot have one type of limitation without having the other. The modern use of the term mathematical logic encompasses (at least) the areas of proof theory (it studies the structure, properties, and limitations of proofs), model theory (it studies the interplay between syntax and meaning – or semantics – by looking at the algebraic structures where formal languages are interpreted), recursion theory (or computability, which studies the properties and limitations of algorithmic processes), and set theory. The fact that the lastmentioned will totally occupy our attention in volume 2 is reflected in the prominence of the term in the title of these lectures. It also reflects a tendency, even today, to think of set theory as a branch in its own right, rather than as an “area” under a wider umbrella.

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Volume 1 is a brief study of the other three areas of logic† mentioned above. This is the point where an author usually apologizes for what has been omitted, blaming space or scope (or competence) limitations. Let me start by outlining what is included: “Standard” phenomena such as completeness, compactness and its startling application to analysis, incompleteness or unprovability (including a complete proof of the second incompleteness theorem), and a fair amount of recursion theory are thoroughly discussed. Recursion theory, or computability, is of interest to a wide range of audiences, including students with main areas of study such as computer science, philosophy, and, of course, mathematical logic. It studies among other things the phenomenon of uncomputability, which is closely related to that of unprovability, as we see in Section I.9. Among the topics that I have deliberately left out are certain algebraic techniques in model theory (such as the method of ultrapowers), formal interpretations of one theory into another,‡ the introduction of “other” logics (modal, higher order, intuitionistic, etc.), and several topics in recursion theory (oracle computability, Turing reducibility, recursive operators, degrees, Post’s theorem in the arithmetic hierarchy, the analytic hierarchy, etc.) – but then, the decision to stop writing within 300 or so pages was firm. On the other hand, the topics included here form a synergistic whole in that I have (largely) included at every stage material that is prerequisite to what follows. The absence of a section on propositional calculus is deliberate, as it does not in my opinion further the understanding of logic in any substantial way, while it delays one’s plunging into what really matters. To compensate, I include all tautologies as “propositional” (or Boolean) logical axioms and present a mini-course on propositional calculus in the exercises of this chapter (I.26–I.41, pp. 193–195), including the completeness and compactness of the calculus. It is inevitable that the language of sets intrudes in this chapter (as it indeed does in all mathematics) and, more importantly, some of the results of (informal) set theory are needed here (especially in our proofs of the completeness and compactness metatheorems). Conversely, formal set theory of volume 2 needs some of the results developed here. This “chicken or egg” phenomenon is often called “bootstrapping” (not to be confused with “circularity” – which it is not§ ), the term suggesting one pulling oneself up by one’s bootstraps.¶ † ‡ § ¶

I trust that the reader will not object to my dropping the qualifier “mathematical” from now on. Although this topic is included in volume 2 (Chapter I), since it is employed in the relative consistency techniques applied there. Only informal, or na¨ıve, set theory notation and results are needed in Chapter I at the meta-level, i.e, outside the formal system that logic is. I am told that Baron M¨unchhausen was the first one to apply this technique, with success.

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3

This is a good place to outline how our story will unfold: First, our objective is to formalize the rules of reasoning in general – as these apply to all mathematics – and develop their properties. In particular, we will study the interaction between formalized rules and their “intended meaning” (semantics), as well as the limitations of these formalized rules: That is, how good (= potent) are they for capturing the informal notions of truth? Secondly, once we have acquired these tools of formalized reasoning, we start behaving (mostly† ) as users of formal logic so that we can discover important theorems of two important mathematical theories: Peano arithmetic (Chapter II) and set theory (volume 2). By formalization (of logic) we understand the faithful representation or simulation of the “reasoning processes” of mathematics in general (pure logic), or of a particular mathematical theory (applied logic: e.g., Peano arithmetic), within an activity that – in principle – is driven exclusively by the form or syntax of mathematical statements, totally ignoring their meaning. We build, describe, and study the properties of this artificial replica of the reasoning processes – the formal theory – within “everyday mathematics” (also called “informal” or “real” mathematics), using the usual abundance of mathematical symbolism, notions, and techniques available to us, augmented by the descriptive power of English (or Greek, or French, or German, or Russian, or . . . , as particular circumstances or geography might dictate). This milieu within which we build, pursue, and study our theories is often called the metatheory, or more generally, metamathematics. The language we speak while at it, this m´elange of mathematics and “natural language”, is the metalanguage. Formalization turns mathematical theories into mathematical objects that we can study. For example, such study may include interesting questions such as “is the continuum hypothesis provable from the axioms of set theory?” or “can we prove the consistency of (axiomatic) Peano arithmetic within Peano arithmetic?”‡ This is analogous to building a “model airplane”, a replica of the real thing, with a view of studying through the replica the properties, power, and limitations of the real thing. But one can also use the formal theory to generate theorems, i.e., discover “truths” in the real domain by simply “running” the simulation that this theoryreplica is.§ Running the simulation “by hand” (rather than using the program † ‡ §

Some tasks in Chapter II of this volume, and some others in volume 2, will be to treat the “theory” at hand as an object of study rather than using it, as a machine, to crank out theorems. By the way, the answer to both these questions is “no” (Cohen (1963) for the first, G¨odel (1938) for the second). The analogy implied in the terminology “running the simulation” is apt. For formal theories such as set theory and Peano arithmetic we can build within real mathematics a so-called “provability

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of the previous footnote) means that you are acting as a “user” of the formal system, a formalist, proving theorems through it. It turns out that once you get the hang of it, it is easier and safer to reason formally than to do so informally. The latter mode often mixes syntax and semantics (meaning), and there is always the danger that the “user” may assign incorrect (i.e., convenient, but not general ) meanings to the symbols that he† manipulates, a phenomenon that has distressed many a mathematics or computer science instructor. “Formalism for the user” is hardly a revolutionary slogan. It was advocated by Hilbert, the founder of formalism, partly as a means of – as he believed‡ – formulating mathematical theories in a manner that allows one to check them (i.e., run “diagnostic tests” on them) for freedom from contradiction,§ but also as the right way to “do” mathematics. By this proposal he hoped to salvage mathematics itself, which, Hilbert felt, was about to be destroyed by the Brouwer school of intuitionist thought. In a way, his program could bridge the gap between the classical and the intuitionist camps, and there is some evidence that Heyting (an influential intuitionist and contemporary of Hilbert) thought that such a rapprochement was possible. After all, since meaning is irrelevant to a formalist, then all that he is doing (in a proof) is shuffling finite sequences of symbols, never having to handle or argue about infinite objects – a good thing, as far as an intuitionist is concerned.¶

† ‡ §

¶

predicate”, that is, a relationP(y, x) which is true of two natural numbers y and x just in case y codes a proof of the formula coded by x. It turns out that P(y, x) has so simple a structure that it is programmable, say in the C programming language. But then we can write a program (also in C) as follows: “Systematically generate all the pairs of numbers (y, x). For each pair generated, if P(y, x) holds, then print the formula coded by x”. Letting this process run for ever, we obtain a listing of all the theorems of Peano arithmetic or set theory! This fact does not induce any insomnia in mathematicians, since this is an extremely impractical way to obtain theorems. By the way, we will see in Chapter II that either set theory or Peano arithmetic is sufficiently strong to formally express a provability predicate, and this leads to the incompletableness phenomenon. In this volume, the terms “he”, “his”, “him”, and their derivatives are by definition gender-neutral. This belief was unfounded, as G¨odel’s incompleteness theorems showed. Hilbert’s metatheory – that is, the “world” or “lab” outside the theory, where the replica is actually manufactured – was finitary. Thus – Hilbert advocated – all this theory building and theory checking ought to be effected by finitary means. This ingredient of his “program” was consistent with peaceful coexistence with the intuitionists. And, alas, this ingredient was the one that – as some writers put it – destroyed Hilbert’s program to found mathematics on his version of formalism. G¨odel’s incompleteness theorems showed that a finitary metatheory is not up to the task. True, a formalist applies classical logic, while an intuitionist applies a different logic where, for example, double negation is not removable. Yet, unlike a Platonist, a Hilbert-style formalist does not believe – or he does not have to disclose to his intuitionist friends that he might believe – that infinite sets exist in the metatheory, as his tools are just finite symbol sequences. To appreciate the tension here, consider this anecdote: It is said that when Kronecker – the father of intuitionism – was informed of Lindemann’s proof (1882) that π is transcendental, while he granted that this was an interesting result, he also dismissed it, suggesting that “π ” – whose decimal expansion is, of

I.1. First Order Languages

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In support of the “formalism for the user” position we must definitely mention the premier paradigm, Bourbaki’s monumental work (1966a), which is a formalization of a huge chunk of mathematics, including set theory, algebra, topology, and theory of integration. This work is strictly for the user of mathematics, not for the metamathematician who studies formal theories. Yet, it is fully formalized, true to the spirit of Hilbert, and it comes in a self-contained package, including a “Chapter 0” on formal logic. More recently, the proposal to employ formal reasoning as a tool has been gaining support in a number of computer science undergraduate curricula, where logic and discrete mathematics are taught in a formalized setting, starting with a rigorous course in the two logical calculi (propositional and predicate), emphasizing the point of view of the user of logic (and mathematics) – hence with an attendant emphasis on “calculating” (i.e., writing and annotating formal) proofs. Pioneering works in this domain are the undergraduate text (1994) and the paper (1995) of Gries and Schneider. I.1. First Order Languages In the most abstract (therefore simplest) manner of describing it, a formalized mathematical theory consists of the following sets of things: A set of basic or primitive symbols, V , used to build symbol sequences (also called strings, or expressions, or words) “over V ”. A set of strings, Wff, over V , called the formulas of the theory. Finally, a subset of Wff, called Thm, the set of theorems of the theory.† Well, this is the extension of a theory, that is, the explicit set of objects in it. How is a theory “given”? In most cases of interest to the mathematician it is given by V and two sets of simple rules: formula-building rules and theorem-building rules. Rules from the first set allow us to build, or generate, Wff from V . The rules of the second set generate Thm from Wff. In short (e.g., Bourbaki (1966b)), a theory consists of an alphabet of primitive symbols, some rules used to generate the “language of the theory” (meaning, essentially, Wff) from these symbols, and some additional rules used to generate the theorems. We expand on this below:

†

course, infinite but not periodic – “does not exist” (see Wilder (1963, p. 193)). We are not to propound the tenets of intuitionism here, but it is fair to state that infinite sets are possible in intuitionistic mathematics as this has later evolved in the hands of Brouwer and his Amsterdam “school”. However, such sets must be (like all sets of intuitionistic mathematics) finitely generated – just as our formal languages and the set of theorems are (the latter provided our axioms are too) – in a sense that may be familiar to some readers who have had a course in “automata and language theory”. See Wilder (1963, p. 234) For a less abstract, but more detailed view of theories see p. 38.

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I.1.1 Remark . What is a “rule”? We run the danger of becoming circular or too pedantic if we overdefine this notion. Intuitively, the rules we have in mind are string manipulation rules, that is, “black boxes” (or functions) that receive string inputs and respond with string outputs. For example, a well-known theorembuilding rule receives as input a formula and a variable, and returns (essentially) the string composed of the symbol ∀, immediately followed by the variable and,
in turn, immediately followed by the formula.† (1) First off, the ( first order) formal language, L, where the theory is “spoken”,‡ is a triple (V , Term, Wff), that is, it has three important components, each of them a set. V is the alphabet or vocabulary of the language. It is the collection of the basic syntactic “bricks” (symbols) that we use to form expressions that are terms (members of Term) or formulas (members of Wff). We will ensure that the processes that build terms or formulas, using the basic building blocks in V , are intuitively algorithmic or “mechanical”. Terms will formally codify “objects”, while formulas will formally codify “statements” about objects. (2) Reasoning in the theory will be the process of discovering true statements about objects – that is, theorems. This discovery journey begins with certain formulas which codify statements that we take for granted (i.e., we accept without “proof” as “basic truths”). Such formulas are the axioms. There are two types of axioms: Special or nonlogical axioms are to describe specific aspects of any specific theory that we might be building. For example, “x + 1 = 0” is a special axiom that contributes towards the characterization of number theory over the natural numbers, N. The other kind of axiom will be found in all theories. It is the kind that is “universally valid”, that is, not theory-specific (for example, “x = x” is such a “universal truth”). For that reason this type of axiom will be called logical. (3) Finally, we will need rules for reasoning, actually called rules of inference. These are rules that allow us to deduce, or derive, a true statement from other statements that we have already established as being true.§ These rules will be chosen to be oblivious to meaning, being only concerned with † ‡ §

This rule is usually called “generalization”. We will soon say what makes a language “first order”. The generous use of the term “true” here is only meant for motivation. “Provable” or “deducible” (formula), or “theorem”, will be the technically precise terminology that we will soon define to replace the term “true statement”.

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form. They will apply to statement “configurations” of certain recognizable forms and will produce (derive) new statements of some corresponding recognizable forms (See Remark I.1.1). I.1.2 Remark. We may think of axioms of either logical or nonlogical type as special cases of rules, that is, rules that receive no input in order to produce an output. In this manner item (2) above is subsumed by item (3), and thus we are faithful to our abstract definition of theory where axioms were not mentioned. An example, outside mathematics, of an inputless rule is the rule invoked when you type date on your computer keyboard. This rule receives no input,
and outputs on your screen the current date. We next look carefully into (first order) formal languages. There are two parts in each first order alphabet. The first, the collection of the logical symbols, is common to all first order languages regardless of which theory is “spoken” in them. We describe this part immediately below. Logical Symbols LS.1. Object or individual variables. An object variable is any one symbol out of the non-ending sequence v0 , v1 , v2 , . . . . In practice – whether we are using logic as a tool or as an object of study – we agree to be sloppy with notation and use, generically, x, y, z, u, v, w with or without subscripts or primes as names of object variables.† This is just a matter of notational convenience. We allow ourselves to write, say, z instead of, say, v1200000000560000009 . Object variables (intuitively) “vary over” (i.e., are allowed to take values that are) the objects that the theory studies (numbers, sets, atoms, lines, points, etc., as the case may be). LS.2. The Boolean or propositional connectives. These are the symbols “¬” and “∨”.‡ They are pronounced not and or respectively. LS.3. The existential quantifier, that is, the symbol “∃”, pronounced exists or for some. LS.4. Brackets, that is, “(” and “)”. LS.5. The equality predicate. This is the symbol “=”, which we use to indicate that objects are “equal”. It is pronounced equals. † ‡

Conventions such as this one are essentially agreements – effected in the metatheory – on how to be sloppy and get away with it. They are offered in the interest of user-friendliness. The quotes are not part of the symbol. They serve to indicate clearly here, in particular in the case of “∨”, what is part of the symbol and what is not (the following period).

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The logical symbols will have a fixed interpretation. In particular, “=” will always be expected to mean equals. The theory-specific part of the alphabet is not fixed, but varies from theory to theory. For example, in set theory we just add the nonlogical (or special) symbols, ∈ and U . The first is a special predicate symbol (or just predicate) of arity 2, the second is a predicate symbol of arity 1.† In number theory we adopt instead the special symbols S (intended meaning: successor, or “ + 1” function), +, ×, 0, 0. We normally use P, Q, R generically, with or without primes or subscripts, to stand for predicate symbols. Note that = is in the logical camp. Also note that theoryspecific formal symbols are possible for predicates, e.g., 0. We normally use f, g, h, generically, with or without primes or subscripts, to stand for function symbols. Note that theoryspecific formal symbols are possible for functions, e.g., +, ×. I.1.3 Remark. (1) We have the option of assuming that each of the logical symbols that we named in LS.1–LS.5 have no further “structure” and that the symbols are, ontologically, identical to their names, that is, they are just these exact signs drawn on paper (or on any equivalent display medium). In this case, changing the symbols, say, ¬ and ∃ to ∼ and E respectively results in a “different” logic, but one that is, trivially, “isomorphic” to the one †

‡

“Arity” is a term mathematicians have made up. It is derived from “ary” of “unary”, “binary”, etc. It denotes the number of arguments needed by a symbol according to the dictates of correct syntax. Function and predicate symbols need arguments. Metasymbols are informal (i.e., outside the formal language) symbols that we use within “everyday” or “real” mathematics – the metatheory – in order to describe, as we are doing here, the formal language.

I.1. First Order Languages

9

we are describing: Anything that we may do in, or say about, one logic trivially translates to an equivalent activity in, or utterance about, the other as long as we systematically carry out the translations of all occurrences of ¬ and ∃ to ∼ and E respectively (or vice versa). An alternative point of view is that the symbol names are not the same as (identical with) the symbols they are naming. Thus, for example, “¬” names the connective we pronounce not, but we do not know (or care) exactly what the nature of this connective is (we only care about how it behaves). Thus, the name “¬” becomes just a typographical expedient and may be replaced by other names that name the same object, not. This point of view gives one flexibility in, for example, deciding how the variable symbols are “implemented”. It often is convenient to think that the entire sequence of variable symbols was built from just two symbols, say, “v” and “|”.† One way to do this is by saying that vi is a name for the symbol sequence‡ “v | . . . | ”
  i|’s

Or, preferably – see (2) below – vi might be a name for the symbol sequence “v | . . . | v”
  i|’s

Regardless of option, vi and v j will name distinct objects if i = j. This is not the case for the metavariables (“abbreviated informal names”) x, y, z, u, v, w. Unless we say so explicitly otherwise, x and y may name the same formal variable, say, v131 . We will mostly abuse language and deliberately confuse names with the symbols they name. For example, we will say, e.g., “let v1007 be an object variable . . . ” rather than “let v1007 name an object variable . . . ”, thus appearing to favour option one. (2) Any two symbols included in the alphabet are distinct. Moreover, if any of them are built from simpler “sub-symbols” – e.g., v0 , v1 , v2 , . . . might really name the strings vv, v|v, v||v, . . . – then none of them is a substring (or subexpression) of any other.§ †

‡ §

We intend these two symbols to be identical to their names. No philosophical or other purpose will be served by allowing “more indirection” here (such as “v names u, which actually names w, which actually is . . . ”). Not including the quotes. What we have stated under (2) are requirements, not metatheorems! That is, they are nothing of the sort that we can prove about our formal language within everyday mathematics.

10

I. Basic Logic

(3) A formal language, just like a “natural” language (such as English or Greek), is “alive” and evolving. The particular type of evolution we have in mind is the one effected by formal definitions. Such definitions continually add nonlogical symbols to the language.† Thus, when we say that, e.g., “∈ and U are the only nonlogical symbols of set theory”, we are telling a small white lie. More accurately, we ought to have said that “∈ and U are the only ‘primitive’ nonlogical symbols of set theory”, for we will add loads of other symbols such as ∪, ω, ∅, ⊂, ⊆. This evolution affects the (formal) language of any theory, not just set theory.
Wait a minute! If formal set theory is “the foundation of all mathematics”, and if, ostensibly, this chapter on logic assists us to found set theory itself, then how come we are employing natural numbers like 1200000000560000009 as subscripts in the names of object variables? How is it permissible to already talk about “sets of symbols” when we are about to found a theory of sets formally? Surely we do not “have”‡ any of these “items” yet, do we? First off, the presence of subscripts such as 1200000000560000009 in v1200000000560000009 is a non-issue. One way to interpret what has been said in the definition is to view the various vi as abbreviated names of the real thing, the latter being strings that employ the symbols v and | as in Remark I.1.3. In this connection saying that vi is “implemented” as v|...|v
 

(1)

i|’s

especially the use of “i” above, is only illustrative, thus totally superfluous. We can say instead that strings of type (1) are the variables which we define as follows without the help of the “natural number i” (this is a variation of how this is done in Bourbaki (1966b) and Hermes (1973)): An “|-calculation” forms a string like this: Write a “|”.§ This is the “current string”. Repeat a finite number of times: Add (i.e., concatenate) one | immediately to the right of the current string. Write this new string (it is now the current string). †

‡ §

This phenomenon will be studied in some detail in what follows. By the way, any additions are made to the nonlogical side of the alphabet. All the logical symbols have been given, once and for all. “Do not have” in the sense of having not formally defined – or proved to exist – or both. Without the quotes. These were placed to exclude the punctuation following.

I.1. First Order Languages

11

Let us call any string that figures in some |-calculation a “|-string”. A variable either is the string vv, or is obtained as the concatenation from left to right of v followed by an |-string, followed by v. All we now need is the ability to generate as many as necessary distinct variables (this is the “non-ending sequence” part of the definition, p. 7): For any two variables we get a new one that is different from either one by forming the string “v, followed by the concatenation of the two |-parts, followed by v”. Similarly if we had three, four, . . . variables. By the way, two strings of | are distinct iff† both occur in the same |-calculation, one, but not both, as the last string. Another, more direct way to interpret what was said about object variables on p. 7 is to take the definition literally, i.e., to suppose that it speaks about the ontology of the variables.‡ Namely, the subscript is just a a string of meaningless symbols taken from the list below: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Again we can pretend that we know nothing about natural numbers, and whenever, e.g., we want a variable other than either of v123 or v321 , we may offer either of v123321 or v321123 as such a new variable. O.K., so we have not used natural numbers in the definition. But we did say “sets” and also “non-ending sequence”, implying the presence of infinite sets! As we have already noted, on one hand we have “real mathematics”, and on the other hand we have syntactic replicas of theories – the formal theories – that we built within real mathematics. Having built a formal theory, we can then choose to use it (acting like formalists) to generate theorems, the latter being codified as symbol sequences (formulas). Thus, the assertion “axiomatic set theory is the foundation of all mathematics” is just a colloquialism proffered in the metatheory that means that “within axiomatic set theory we can construct the known sets of mathematics, such as the reals R and the complex numbers C, and moreover we can simulate what we informally do whenever we are working in real or complex analysis, algebra, topology, theory of measure and integration, functional analysis, etc., etc.” There is no circularity here, but simply an empirical boastful observation in the metatheory of what our simulator can do. Moreover, our metatheory does † ‡

If and only if. Why not just say exactly what a definition is meant to say rather than leave it up to interpretation? One certainly could, as in Bourbaki (1966b), make the ontology of variables crystal-clear right in the definition. Instead, we have followed the custom of more recent writings and given the definition in a quasi-sloppy manner that leaves the ontology of variables as a matter for speculation. This gives one the excuse to write footnotes like this one and remarks like I.1.3.

12

I. Basic Logic

have sets and all sorts of other mathematical objects. In principle we can use any among those towards building or discussing the simulator, the formal theory. Thus, the question is not whether we can use sets, or natural numbers, in our definitions, but whether restrictions apply. For example, can we use infinite sets? If we are Platonists, then we have available in the metatheory all sorts of sets, including infinite sets, in particular the set of all natural numbers. We can use any of these items, speak about them, etc., as we please, when we are describing or building the formal theory within our metatheory. Now, if we are not Platonists, then our “real” mathematical world is much more restricted. In one extreme, we have no infinite sets.† We can still manage to define our formal language! After all, the “nonending” sequence of object variables v0 , v1 , v2 , . . . can be finitely generated in at least two different ways, as we have already seen. Thus we can explain (to a true formalist or finitist) that “non-ending sequence” was an unfortunate slip of the tongue, and that we really meant to give a procedure of how to generate on demand a new object variable, different from whatever ones we may already have. Two parting comments are in order: One, we have been somewhat selective in the use of the term “metavariable”. We have called x, x , y metavariables, but have implied that the vi are formal variables, even if they are just names of formal objects such that we do not know or do not care what they look like. Well, strictly speaking the abbreviations vi are also metavariables, but they are endowed with a property that the “generic” metavariables like x, y, z do not have: Distinct vi names denote distinct object variables (cf. I.1.3). Two, we should clarify that a formal theory, when used (i.e., the simulator is being “run”) is a generator of strings, not a decider or “parser”. Thus, it can generate any of the following: variables (if these are given by procedures), formulas and terms (to be defined), or theorems (to be defined). Decision issues, no matter how trivial, the system is not built to handle. These belong to the metatheory. In particular, the theory does not see whatever numbers or strings (like 12005) may be hidden in a variable name (such as v12005 ). Examples of decision questions: Is this string a term or a formula or a variable (finitely generated as above)? All these questions are “easy”. They are algorithmically decidable in the metatheory. Or, is this formula a theorem? This is †

A finitist – and don’t forget that Hilbert-style metatheory was finitary, ostensibly for political reasons – will let you have as many integers as you like in one serving, as long as the serving is finite. If you ask for more, you can have more, but never the set of all integers or an infinite subset thereof.

I.1. First Order Languages

13

algorithmically undecidable in the metatheory if it is a question about Peano arithmetic or set theory. I.1.4 Definition (Terminology about Strings). A symbol sequence or expression (or string) that is formed by using symbols exclusively out of a given set† M is called a string over the set, or alphabet, M. If A and B denote strings (say, over M), then the symbol A ∗ B, or more simply AB, denotes the symbol sequence obtained by listing first the symbols of A in the given left to right sequence, immediately followed by the symbols of B in the given left to right sequence. We say that AB is (more properly, denotes or names) the concatenation of the strings A and B in that order. We denote the fact that the strings (named) C and D are identical sequences (but we just say that they are equal) by writing C ≡ D. The symbol ≡ denotes the negation of the string equality symbol ≡. Thus, if # and ? are (we do mean “are”) symbols from an alphabet, then #?? ≡ #?? but

#? ≡ #??

We can also employ ≡ in contexts such as “let A ≡ ##?”, where we give the name A to the string ##?.‡ In this book the symbol ≡ will be exclusively used in the metatheory for equality of strings over some set M. The symbol λ normally denotes the empty string, and we postulate for it the following behaviour: A ≡ Aλ ≡ λA

for all strings A

We say that A occurs in B, or is a substring of B, iff there are strings C and D such that B ≡ CAD. For example, “(” occurs four times in the (explicit) string “¬(()∨)((”, at positions 2, 3, 7, 8. Each time this happens we have an occurrence of “(” in “¬(()∨)((”. If C ≡ λ, we say that A is a prefix of B. If moreover D ≡ λ, then we say
that A is a proper prefix of B. † ‡

A set that supplies symbols to be used in building strings is not special. It is just a set. However, it often has a special name: “alphabet”. Punctuation such as “.” is not part of the string. One often avoids such footnotes by enclosing strings that are explicitly written as symbol sequences inside quotes. For example, if A stands for the string #, one writes A ≡ “#”. Note that we must not write “A”, unless we mean a string whose only symbol is A.

14

I. Basic Logic

I.1.5 Definition (Terms). The set of terms, Term, is the smallest set of strings over the alphabet V with the following two properties: (1) All of the items in LS.1 or NLS.1 (x, y, z, a, b, c, etc.) are included. (2) If f is a function† of arity n and t1 , t2 , . . . , tn are included, then so is the string “ f t1 t2 . . . tn ”. The symbols t, s, and u, with or without subscripts or primes, will denote arbitrary terms. Since we are using them in the metalanguage to “vary over” terms, we naturally call them metavariables. They also serve – as variables – towards the definition (this one) of the syntax of terms. For this reason they are also called syntactic variables.
I.1.6 Remark. (1) We often abuse notation and write f (t1 , . . . , tn ) instead of f t1 . . . tn . (2) Definition I.1.5 is an inductive definition.‡ It defines a more or less “complicated” term by assuming that we already know what “simpler” terms look like. This is a standard technique employed in real mathematics. We will have the opportunity to say more about such inductive definitions – and their appropriateness – in a -comment later on. (3) We relate this particular manner of defining terms to our working definition of a theory (given on p. 6 immediately before Remark I.1.1 in terms of “rules” of formation). Item (2) in I.1.5 essentially says that we build new terms (from old ones) by applying the following general rule: Pick an arbitrary function symbol, say f . This has a specific formation rule associated with it that, for the appropriate number, n, of an already existing ordered list of terms, t1 , . . . , tn , will build the new term consisting of f , immediately followed by the ordered list of the given terms. To be specific, suppose we are working in the language of number theory. There is a function symbol + available there. The rule associated with + builds the new term +ts for any prior obtained terms t and s. For example, +v1 v13 and +v121 + v1 v13 are well-formed terms. We normally write terms of number theory in “infix” notation,§ i.e., t + s, v1 + v13 and v121 + (v1 + v13 ) (note the intrusion of brackets, to indicate sequencing in the application of +). † ‡

§

We will omit from now on the qualification “symbol” from terminology such as “function symbol”, “constant symbol”, “predicate symbol”. Some mathematicians will absolutely insist that we call this a recursive definition and reserve the term “induction” for “induction proofs”. This is seen to be unwarranted hair splitting if we consider that Bourbaki (1966b) calls induction proofs “d´emonstrations par r´ecurrence”. We will be less dogmatic: Either name is all right. Function symbol placed between the arguments.

I.1. First Order Languages

15

A by-product of what we have just described is that the arity of a function symbol f is whatever number of terms the associated rule will require as input. (4) A crucial word used in I.1.5 (which recurs in all inductive definitions) is “smallest”. It means “least inclusive” (set). For example, we may easily think of a set of strings that satisfies both conditions of the above definition, but which is not “smallest” by virtue of having additional elements, such as the string “¬¬(”. Pause. Why is “¬¬(” not in the smallest set as defined above, and therefore not a term? The reader may wish to ponder further on the import of the qualification “smallest” by considering the familiar (similar) example of N, the set of natural numbers. The principle of induction in N ensures that this set is the smallest with the properties: (i) 0 is included, and (ii) if n is included, then so is n + 1. By contrast, all of Z (set of integers), Q (set of rational numbers), R (set of real
numbers) satisfy (i) and (ii), but they are clearly not the “smallest” such. I.1.7 Definition (Atomic Formulas). The set of atomic formulas, Af, contains precisely: (1) The strings t = s for every possible choice of terms t, s. (2) The strings Pt1 t2 . . . tn for every possible choice of n-ary predicates P (for all choices of n > 0) and all possible choices of terms t1 , t2 , . . . , tn .
We often abuse notation and write P(t1 , . . . , tn ) instead of Pt1 . . . tn . I.1.8 Definition (Well-Formed Formulas). The set of well-formed formulas, Wff, is the smallest set of strings or expressions over the alphabet V with the following properties: (a) All the members of Af are included. (b) If A and B denote strings (over V ) that are included, then (A ∨ B ) and (¬A) are also included. (c) If A is† a string that is included and x is any object variable (which may or may not occur (as a substring) in the string A), then the string ((∃x)A ) is
also included. We say that A is the scope of (∃x). †

Denotes!

16

I. Basic Logic

I.1.9 Remark. (1) The above is yet another inductive definition. Its statement (in the metalanguage) is facilitated by the use of so-called syntactic, or meta, variables – A and B – used as names for arbitrary (indeterminate) formulas. In general, we will let calligraphic capital letters A, B , C , D , E , F , G (with or without primes or subscripts) be names for well-formed formulas, or just formulas, as we often say. The definition of Wff given above is standard. In particular, it permits well-formed formulas such as ((∃x)((∃x)x = 0)) in the interest of making the formation rules “context-free”.† (2) The rules of syntax just given do not allow us to write things such as ∃ f or ∃P where f and P are function and predicate symbols respectively. That quantification is deliberately restricted to act solely on object variables makes the language first order. (3) We have already indicated in Remark I.1.6 where the arities (of function and predicate symbols) come from (Definitions I.1.5 and I.1.7 referred to them). These are numbers that are implicit (“hardwired”) with the formation rules for terms and atomic formulas. Each function and each predicate symbol (e.g., +, ×, ∈, n, since we can lengthen a derivation arbitrarily by adding I -elements to it. I.2.5 Remark. The following metatheorem shows that there is a way to “construct” Cl(I , R) iteratively, i.e., one element at a time by repeated application of the rules. This result shows definitively that our inductive definitions of terms (I.1.5) and well-formed formulas (I.1.8) fully conform with our working definition of theory, as an alphabet and a set of rules that are used to build formulas and
theorems (p. 5). I.2.6 Metatheorem. Cl(I , R) = {x : x is (I , R)-derivable within some number of steps, n} Proof. For notational convenience let us write T = {x : x is (I , R)-derivable within some number of steps, n}. As we know from elementary na¨ıve set theory, we need to show here both Cl(I , R) ⊆ T and Cl(I , R) ⊇ T to settle the claim. (⊆) We do induction on Cl(I , R) (using I.2.3). Now I ⊆ T , since every member of I is derivable in n = 1 step. (Why?) Also, T is closed under every Q in R. Indeed, let such an (r + 1)-ary Q be chosen, and assume Q(a1 , . . . , ar , b) †

This “or” is inclusive: (1), or (2), or both.

(i)

22

I. Basic Logic

and {a1 , . . . , ar } ⊆ T . Thus, each ai has a (I , R)-derivation. Concatenate all these derivations: . . . , a1 , . . . , a2 , . . . , . . . , ar The above is a derivation (why?). But then, so is . . . , a1 , . . . , a2 , . . . , . . . , ar , b by (i). Thus, b ∈ T . (⊇) We argue this – that is, “if d ∈ T , then d ∈ Cl(I , R)” – by induction on the number of steps, n, in which d is derivable. For n = 1 we have d ∈ I and we are done, since I ⊆ Cl(I , R). Let us make the induction hypothesis (I.H.) that for derivations of ≤ n steps the claim is true. Let then d be derivable within n + 1 steps. Thus, there is a derivation a1 , . . . , an , d. Now, if d ∈ I , we are done as above (is this a “real case”?) If on the other hand Q(a j1 , . . . , a jr , d), then for i = 1, . . . , r we have a ji ∈ Cl(I , R) by the
I.H.; hence d ∈ Cl(I , R), since the closure is closed under all Q ∈ R. I.2.7 Example. One can see now that N = Cl(I , R), where I = {0} and R contains just the relation y = x + 1 (input x, output y). Similarly, Z, the set of all integers, is Cl(I , R), where I = {0} and R contains just the relations y = x + 1 and y = x − 1 (input x, output y). For the latter, the inclusion Cl(I , R) ⊆ Z is trivial (by I.2.3). For ⊇ we easily see that any n ∈ Z has a (I , R)-derivation (and then we are done by I.2.6). For example, if n > 0, then 0, 1, 2, . . . , n is a derivation, while if n < 0, then 0, −1, −2, . . . , n is one. If n = 0, then the one-term sequence 0 is a derivation. Another interesting closure is obtained by I = {3} and the two relations
z = x + y and z = x − y. This is the set {3k : k ∈ Z} (see Exercise I.1). Pause. So, taking the first sentence of I.2.7 one step further, we note that we have just proved the induction principle for N, for that is exactly what the “equation” N = Cl(I , R) says (by I.2.3). Do you agree? There is another way to view the iterative construction of Cl(I , R): The set is constructed in stages. Below we are using some more notation borrowed from informal set theory. For any sets A and B we write A ∪ B to indicate the set union, which consists of all the members found in A or B or in both. More generally, if we have a lot of sets, X 0 , X 1 , X 2 , . . . , that is, one X i for every integer i ≥ 0 – which we denote by the compact notation (X i )i ≥ 0 – then we may wish to form a set that includes all the objects found as members all over the X i , that is (using inclusive, or logical, “or”s below), form {x : x ∈ X 0 or x ∈ X 1 or . . . }

I.2. A Digression into the Metatheory

23

or, more elegantly and precisely, {x : for some i ≥ 0, x ∈ X i } The latter is called the union of the sequence (X i )i≥0 and is often denoted by   X or Xi i i≥0 i≥0

Correspondingly, we write  i≤n

Xi

or



Xi

i≤n

if we only want to take a finite union, also indicated clumsily as X 0 ∪ . . . ∪ X n . I.2.8 Definition (Stages). In connection with Cl(I , R) we define the sequence of sets (X i )i ≥ 0 by induction on n, as follows: X0 = I  X n+1 =



 Xi

i≤n



∪ b : for some Q ∈ R and some an in



X i , Q( an , b)

i ≤n

That is, to form X n+1 we append to i ≤ n X i all the outputs of all the relations

in R acting on all possible inputs, the latter taken from i≤n X i . We say that X i is built at stage i, from initial objects I and rule-set R.
In words, at stage 0 we are given the initial objects (X 0 = I ). At stage 1 we apply all possible relations to all possible objects that we have so far – they form the set X 0 – and build the 1st stage set, X 1 , by appending the outputs to what we have so far. At stage 2 we apply all possible relations to all possible objects that we have so far – they form the set X 0 ∪ X 1 – and build the 2nd stage set, X 2 , by appending the outputs to what we have so far. And so on. When we work in the metatheory, we take for granted that we can have simple inductive definitions on the natural numbers. The reader is familiar with several such definitions, e.g., a0 = 1 (for a = 0 throughout) a = a · an n+1

24

I. Basic Logic

We will (meta)prove a general theorem on the feasibility of recursive definitions later on (I.2.13). The following theorem connects stages and closures. I.2.9 Metatheorem. With the X i as in I.2.8,  Cl(I , R) = Xi i≥0

Proof. (⊆) We do induction on Cl(I , R). For the basis, I = X 0 ⊆ i≥0 X i .

an , b) hold, for some We show that i≥0 X i is R-closed. Let Q ∈ R and Q(

an in i≥0 X i . Thus, by definition of union, there are integers j1 , j2 , . . . , jn

such that ai ∈ X ji , i = 1, . . . , n. If k = max{ j1 , . . . , jn }, then an is in i≤k X i ;

hence b ∈ X k + 1 ⊆ i≥0 X i . (⊇) It suffices to prove that X n ⊆ Cl(I , R), a fact we can prove by induction on n. For n = 0 it holds by I.2.2. As an I.H. we assume the claim for all n ≤ k.

The case for k + 1: X k + 1 is the union of two sets. One is i≤k X i . This is a subset of Cl(I , R) by the I.H. The other is 

b : for some Q ∈ R and some a in X i , Q( a , b) i≤k

This too is a subset of Cl(I , R), by the preceding observation and the fact that
Cl(I , R) is R-closed. Worth Saying. An inductively defined set can be built by stages. I.2.10 Definition (Immediate Predecessors, Ambiguity). If d ∈ Cl(I , R) and for some Q and a1 , . . . , ar it is the case that Q(a1 , . . . , ar , d), then the a1 , . . . , ar are immediate Q-predecessors of d, or just immediate predecessors if Q is understood; for short, i.p. A pair (I , R) is called ambiguous if some d ∈ Cl(I , R) satisfies any (or all) of the following conditions: (i) It has two (or more) distinct sets of immediate P-predecessors for some rule P. (ii) It has both immediate P-predecessors and immediate Q-predecessors, for P = Q. (iii) It is a member of I , yet it has immediate predecessors. If (I , R) is not ambiguous, then it is unambiguous.




I.2. A Digression into the Metatheory

25

I.2.11 Example. The pair ({00,0}, {Q}), where Q(x, y, z) holds iff z = x y (where “x y” denotes the concatenation of the strings x and y, in that order), is ambiguous. For example, 0000 has the two immediate predecessor sets {00,00} and {0,000}. Moreover, while 00 is an initial object, it does have immediate predecessors, namely, the set {0,0} (or, what amounts to the same thing, {0}).
I.2.12 Example. The pair (I , R), where I = {3} and R consists of z = x + y and z = x − y, is ambiguous. Even 3 has (infinitely many) distinct sets of i.p. (e.g., any {a, b} such that a + b = 3, or a − b = 3). The pairs that effect the definition of Term (I.1.5) and Wff (I.1.8) are un
ambiguous (see Exercises I.2 and I.3). I.2.13 Metatheorem (Definition by Recursion). Let (I , R) be unambiguous and Cl(I , R) ⊆ A, where A is some set. Let also Y be a set, and† h : I → Y and gQ , for each Q ∈ R, be given functions. For any (r +1)-ary Q, an input for the function gQ is a sequence a, b1 , . . . , br  where a is in A and the b1 , . . . , br are all in Y . All the gQ yield outputs in Y . Under these assumptions, there is a unique function f : Cl(I , R) → Y such that  y = h(x) and x ∈ I    or, for some Q ∈ R, (1) y = f (x) iff  y = gQ (x, o1 , . . . , or ) and Q(a1 , . . . , ar , x) holds,   where oi = f (ai ), for i = 1, . . . , r The reader may wish to skip the proof on first reading.  by‡ Proof. Existence part. For each (r + 1)-ary Q ∈ R, define Q  1 , o1 , . . . , ar , or , b, gQ (b, o1 , . . . , or )) Q(a

iff

Q(a1 , . . . , ar , b)

(2)

 is effected for all possible For any a1 , . . . , ar , b, the above definition of Q choices of o1 , . . . , or such that gQ (b, o1 , . . . , or ) is defined.  to form a set of rules R . Collect now all the Q Let also I = {x, h(x) : x ∈ I }. † ‡

The notation f : A → B is common in informal (and formal) mathematics. It denotes a function f that receives “inputs” from the set A and yields “outputs” in the set B. For a relation Q, writing just “Q(a1 , . . . , ar , b)” is equivalent to writing “Q(a1 , . . . , ar , b) holds”.

26

I. Basic Logic

 ) is a 2-ary relation that for every We will verify that the set F = Cl(I, R input yields at most one output, and therefore is a function. For such a relation it is customary to write, letting the context fend off the obvious ambiguity in the use of the letter F, y = F(x)

iff

F(x, y)

(∗)

We will further verify that replacing f in (1) above by F results in a valid equivalence (the “iff” holds). That is, F satisfies (1). (a) We establish that F is a relation composed of pairs x, y (x is input, y is output), where x ∈ Cl(I , R) and y ∈ Y . This follows easily by induction on F (I.2.3), since I ⊆ F, and the property (of “containing such pairs”)  (recall that the gQ yield outputs in Y ). propagates with each Q (b) We next show that “if x, y ∈ F and x, z ∈ F, then y = z”, that is, F is “single-valued” or “well-defined”, in short, it is a function. We again employ induction on F, thinking of the quoted statement as a “property” of the pair x, y: Suppose that x, y ∈ I, and let also x, z ∈ F.  1 , o1 , . . . , ar , or , x, z), where By I.2.6, x, z ∈ I, or Q(a  and Q(a1 , . . . , ar , x) and z = g Q (x, o1 , . . . , or ), for some (r + 1)-ary Q a1 , o1 , . . . , ar , or  in F. The right hand side of the italicized “or” cannot hold for an unambiguous (I , R), since x cannot have i.p. Thus x, z ∈ I; hence y = h(x) = z.  let To prove that the property propagates with each Q,  1 , o1 , . . . , ar , or , x, y) Q(a but also

       P b1 , o1 , . . . , bl , ol , x, z

where Q(a1 , . . . , ar , x), P(b1 , . . . , bl , x), and y = gQ (x, o1 , . . . , or )

and

  z = gP x, o1 , . . . , ol

(3)

 = P),  r = l, Since (I , R) is unambiguous, we have Q = P (hence also Q and ai = bi for i = 1, . . . , r . By I.H., oi = oi for i = 1, . . . , r ; hence y = z by (3).  ) to prove: (c) Finally, we show that F satisfies (1). We do induction on Cl(I, R (←) If x ∈ I and y = h(x), then F(x, y) (i.e., y = F(x) in the alternative notation (∗)), since I ⊆ F. Let next y = gQ (x, o1 , . . . , or ) and Q(a1 , . . . , ar , x), where also F(ai , oi ), for i = 1, . . . , r . By (2),  1 , o1 , . . . , ar , or , x, g Q (x, o1 , . . . , or )); thus – F being closed Q(a

I.2. A Digression into the Metatheory

27

 – F(x, g Q (b, o1 , . . . , or )) holds; in short, F(x, y) under all the rules in R or y = F(x). (→) Now we assume that F(x, y) holds and we want to infer the right hand side (of iff ) in (1). We employ Metatheorem I.2.6. Case 1. Let x, y be F-derivable† in n = 1 step. Then x, y ∈ I. Thus y = h(x). Case 2. Suppose next that x, y is F-derivable within n + 1 steps, namely, we have a derivation x1 , y1 , x2 , y2 , . . . , xn , yn , x, y

(4)

 1 , o1 , . . . , ar , or , x, y) and Q(a1 , . . . , ar , x) (see (2)), where Q(a and each of a1 , o1 , . . . , ar , or  appears in the above derivation, to the left of x, y. This entails (by (2)) that y = g Q (x, o1 , . . . , or ). Since the ai , oi  appear in (4), F(ai , oi ) holds, for i = 1, . . . , r . Thus, x, y satisfies the right hand side of iff in (1), once more. Uniqueness part. Let the function K also satisfy (1). We show, by induction on Cl(I , R), that For all x ∈ Cl(I , R) and all y ∈ Y,

y = F(x) iff y = K (x)

(5)

(→) Let x ∈ I , and y = F(x). By lack of ambiguity, the case conditions of (1) are mutually exclusive. Thus, it must be that y = h(x). But then, y = K (x) as well, since K satisfies (1) too. Let now Q(a1 , . . . , ar , x) and y = F(x). By (1), there are (unique, as we now know) o1 , . . . , or such that oi = F(ai ) for i = 1, . . . , r , and y = gQ (x, o1 , . . . , or ). By the I.H., oi = K (ai ). But then (1) yields y = K (x) as well (since K satisfies (1)). (←)

Just interchange the letters F and K in the above argument.




The above clearly is valid for functions h and gQ that may fail to be defined everywhere in their “natural” input sets. To be able to have this degree of generality without having to state additional definitions (such as left fields, right fields, partial functions, total functions, nontotal functions, Kleene “weak equality”), we have stated the recurrence (1) the way we did (to keep an eye on both the input and output side of things) rather than the “usual”  h(x) if x ∈ I f (x) = gQ (x, f (a1 ), . . . , f (ar )) if Q(a1 , . . . , ar , x) holds †

 )-derivable. Cl(I, R

28

I. Basic Logic

Of course, if all the gQ and h are defined everywhere on their input sets (i.e., they are “total”), then f is defined everywhere on Cl(I , R) (see Exercise I.4). I.3. Axioms and Rules of Inference Now that we have our language, L, we will embark on using it to formally effect deductions. These deductions start at the axioms. Deductions employ “acceptable” purely syntactic – i.e., based on form, not on meaning – rules that allow us to write a formula down (to deduce it) solely because certain other formulas that are syntactically related to it were already deduced (i.e., already written down). These string-manipulation rules are called rules of inference. We describe in this section the axioms and the rules of inference that we will accept into our logical calculus and that are common to all theories. We start with a precise definition of tautologies in our first order language L. I.3.1 Definition (Prime Formulas in Wff. Propositional Variables). A formula A ∈ Wff is a prime formula or a propositional variable iff it is either of Pri1. atomic, Pri2. a formula of the form ((∃x)A). We use the lowercase letters p, q, r (with or without subscripts or primes) to denote arbitrary prime formulas (propositional variables) of our language.
That is, a prime formula has either no propositional connectives, or if it does, it hides them inside the scope of (∃x). We may think of a propositional variable as a “blob” that a myopic being makes out of a formula described in I.3.1. The same being will see an arbitrary well-formed formula as a bunch of blobs, brackets, and Boolean connectives (¬, ∨), “correctly connected” as stipulated below.† I.3.2 Definition (Propositional Formulas). The set of propositional formulas over V , denoted here by Prop, is the smallest set such that: (1) Every propositional variable (over V ) is in Prop. (2) If A and B are in Prop, then so are (¬A) and (A ∨ B ). We use the lowercase letters p, q, r (with or without subscripts or primes) to denote arbitrary prime formulas (propositional variables) of our language.
†

Interestingly, our myope can see the brackets and the Boolean connectives.

I.3. Axioms and Rules of Inference

29

I.3.3 Metatheorem. Prop = Wff. Proof. (⊆) We do induction on Prop. Every item in I.3.2(1) is in Wff. Wff satisfies I.3.2(2) (see I.1.8(b)). Done. (⊇) We do induction on Wff. Every item in I.1.8(a) is a propositional variable (over V ), and hence is in Prop. Prop trivially satisfies I.1.8(b). It also satisfies I.1.8(c), for if A is in Prop, then it is in Wff by the ⊆-direction, above. Then, by I.3.1, ((∃x)A ) is a propo
sitional variable and hence in Prop. We are done once more. I.3.4 Definition (Propositional Valuations). We can arbitrarily assign a value of 0 or 1 to every A in Wff (or Prop) as follows: (1) We fix an assignment of 0 or 1 to every prime formula. We can think of this as an arbitrary but fixed function v : {all prime formulas over L} → {0, 1} in the metatheory. (2) We define by recursion an extension of v, denoted by v¯ : v¯ ((¬A)) = 1 − v¯ (A) v¯ ((A ∨ B )) = v¯ (A) · v¯ (B ) where “·” above denotes number multiplication. We call, traditionally, the values 0 and 1 by the names “true” and “false” respectively, and write t and f respectively. We also call a valuation v a truth (value) assignment. We use the jargon “A takes the truth value t (respectively, f) under a valuation
v” to mean “¯v (A) = 0 (respectively, v¯ (A) = 1)”. The above inductive definition of v¯ relies on the fact that Definition I.3.2 of Prop is unambiguous (I.2.10, p. 24), or that a propositional formula is uniquely readable (or parsable) (see Exercises I.6 and I.7). It employs the metatheorem on recursive definitions (I.2.13). The reader may think that all this about unique readability is just an annoying quibble. Actually it can be a matter of life and death. The ancient Oracle of Delphi had the nasty habit of issuing ambiguous – not uniquely readable, that is – pronouncements. One famous such pronouncement, rendered in English, went like this: “You will go you will return not dying in the war”.† Given that ancient Greeks did not use punctuation, the above has two diametrically opposite meanings depending on whether you put a comma before or after “not”. †

The original was “I ξ εις αϕιξ εις oυ θνηξ εις εν π oλεµ ω”. ι

30

I. Basic Logic

The situation with formulas in Prop would have been as disastrous in the absence of brackets – which serve as punctuation – because unique readability would not be guaranteed: For example, for three distinct prime formulas p, q, r we could find a v such that v¯ ( p → q → r ) is different depending on whether we meant to insert brackets around “ p → q” or around “q → r ” (can you find such a v?). I.3.5 Remark (Truth Tables). Definition I.3.4 is often given in terms of truthfunctions. For example, we could have defined (in the metatheory, of course) the function F¬ : {t, f} → {t, f} by  t if x = f F¬ (x) = f if x = t We could then say that v¯ ((¬A )) = F¬ (¯v (A)). One can similarly take care of all the connectives (∨ and all the abbreviations) with the help of truth functions F∨ , F∧ , F→ , F↔ . These functions are conveniently given via so-called truthtables as indicated below:

x

y

F¬ (x)

F∨ (x, y)

F∧ (x, y)

F→ (x, y)

F↔ (x, y)

f f t t

f t f t

t t f f

f t t t

f f f t

t t f t

t f f t




I.3.6 Definition (Tautologies, Satisfiable Formulas, Unsatisfiable Formulas in Wff). A formula A ∈ Wff (equivalently, in Prop) is a tautology iff for all valuations v one has v¯ (A) = t. We call the set of all tautologies, as defined here, Taut. The symbol |=Taut A says “A is in Taut”. A formula A ∈ Wff (equivalently, in Prop) is satisfiable iff for some valuation v one has v¯ (A ) = t. We say that v satisfies A. A set of formulas  is satisfiable iff for some valuation v, one has v¯ (A) = t for every A in . We say that v satisfies . A formula A ∈ Wff (equivalently, in Prop) is unsatisfiable iff for all valuations v one has v¯ (A) = f. A set of formulas  is unsatisfiable iff for all valuations v one has v¯ (A) = f for some A in .


I.3. Axioms and Rules of Inference

31

I.3.7 Definition (Tautologically Implies, for Formulas in Wff ). Let A and  be respectively any formula and any set of formulas (over L). The symbol |=Taut A, pronounced “ tautologically implies A”, means
that every truth assignment v that satisfies  also satisfies A. “Satisfiable” and “unsatisfiable” are terms introduced here in the propositional or Boolean sense. These terms have a more complicated meaning when we decide to “see” the object variables and quantifiers that occur in formulas. We have at once I.3.8 Lemma. † |=Taut A iff ∪{¬A} is unsatisfiable (in the propositional sense). If = ∅ then |=Taut A says just |=Taut A, since the hypothesis “every truth assignment v that satisfies ”, in the definition above, is vacuously satisfied. For that reason we almost never write ∅ |=Taut A and write instead |=Taut A. I.3.9 Exercise. For any formula A and any two valuations v and v , v¯ (A) = v¯ (A) if v and v agree on all the propositional variables that occur in A. In the same manner, |=Taut A is oblivious to v-variations that do not affect
the variables that occur in  and A (see Exercise I.8). Before presenting the axioms, we need to introduce the concept of substitution. I.3.10 Tentative Definition (Substitutions of Terms). Let A be a formula, x an (object) variable, and t a term. A[x ← t] denotes the result of “replacing” all free occurrences of x in A by the term t, provided no variable of t was “captured” (by a quantifier) during substitution. †

The word “lemma” has Greek origin, “ληµµα”, ´ plural “lemmata” (some people say “lemmas”) from “ληµµατ ´ α”. It derives from the verb “λαµβ ανω” ´ (to take) and thus means “taken thing”. In mathematical reasoning a lemma is a provable auxiliary statement that is taken and used as a stepping stone in lengthy mathematical arguments – invoked therein by name, as in “ . . . by Lemma such and such . . . ” – much as “subroutines” (or “procedures”) are taken and used as auxiliary stepping stones to elucidate lengthy computer programs. Thus our purpose in having lemmata is to shorten proofs by breaking them up into modules.

32

I. Basic Logic

If the proviso is valid, then we say that “t is substitutable for x (in A)”, or that “t is free for x (in A)”. If the proviso is not valid, then the substitution is
undefined. I.3.11 Remark. There are a number of issues about Definition I.3.10 that need discussion or clarification. Reasonable people will be satisfied with the above definition “as is”. However, there are some obscure points (enclosd in quotation marks above). (1) What is this about “capture”? Well, suppose that A ≡ (∃x)¬x = y. Let t ≡ x.† Then A[y ← t] ≡ (∃x)¬x = x, which says something altogether different than the original. Intuitively, this is unexpected (and undesirable): A codes a statement about the free variable y, i.e., a statement about all objects which could be “values” (or meanings) of y. One would have expected that, in particular, A[y ← x] – if the substitution were allowed – would make this very same statement about the values of x. It does not.‡ What happened is that x was captured by the quantifier upon substitution, thus distorting A’s original meaning. (2) Are we sure that the term “replace” is mathematically precise? (3) Is A[x ← t] always a formula, if A is? A re-visitation of I.3.10 via an inductive definition (by induction on terms and formulas) settles (1)–(3) at once (in particular, the informal terms “replace” and “capture” do not appear in the inductive definition). We define (again) the symbol A[x ← t], for any formula A, variable x, and term t, this time by induction on terms and formulas: First off, let us define s[x ← t], where s is also a term, by cases:   t if s ≡ x     a if s ≡ a, a constant  s[x ← t] ≡ (symbol)    y if s ≡ y, a variable ≡ x    f r [x ← t]r [x ← t] . . . r [x ← t] if s ≡ f r . . . r 1 2 n 1 n Pause. Is s[x ← t] always a term? That this is so follows directly by induction on terms, using the definition by cases above and the I.H. that each of ri [x ← t], i = 1, . . . , n, is a term. † ‡

Recall that in I.1.4 (p. 13) we defined the symbol “≡” to be equality on strings. The original says that for any object y there is an object that is different from it; A[y ← x] says that there is an object that is different from itself.

I.3. Axioms and Rules of Inference

33

We turn now to formulas. The symbols P, r, s (with or without subscripts) below denote a predicate of arity n, a term, and a term (respectively):  s[x ← t] = r [x ← t]      Pr1 [x ← t]r2 [x ← t] . . .     rn [x ← t]    (B [x ← t] ∨ C [x ← t]) A[x ← t] ≡  (¬(B [x ← t]))     A     ((∃y)(B [x ← t]))   

if A ≡ s = r if A ≡ Pr1 . . . rn if A ≡ (B ∨ C ) if A ≡ (¬B ) if A ≡ ((∃y)B ) and y ≡ x if A ≡ ((∃y)B ) and y ≡ x and y does not occur in t

In all cases above, the left hand side is defined iff the right hand side is. Pause. We have eliminated “replaces” and “captured”. But is A[x← t] a formula (whenever it is defined)? (See Exercise I.9)
I.3.12 Definition (Simultaneous Substitution). The symbol A[y1 , . . . , yr ← t1 , . . . , tr ] or, equivalently, A[yr ← tr ] – where yr is an abbreviation of y1 , . . . , yr – denotes simultaneous substitution of the terms t1 , . . . , tr into the variables y1 , . . . , yr in the following sense: Let zr be variables that do not occur at all (either as free or bound) in any of A, tr . Then A[yr ← tr ] is short for A[y1 ← z 1 ] . . . [yr ← zr ][z 1 ← t1 ] . . . [zr ← tr ]

(1)


Exercise I.10 shows that we obtain the same string in (1) above, regardless of our choice of new variables zr . More Conventions. The symbol [x ← t] lies in the metalanguage. This metasymbol has the highest priority, so that, e.g., A ∨ B [x ← t] means A ∨ (B [x ← t]), (∃x)B [x ← t] means (∃x)(B [x ← t]), etc. The reader is reminded about the conventions regarding the metanotations A[xr ] and A(xr ) (see I.1.11). In the context of those notations, if t1 , . . . , tr are terms, the symbol A[t1 , . . . , tr ] abbreviates A[yr ← tr ]. We are ready to introduce the (logical) axioms and rules of inference.

34

I. Basic Logic

Schemata.† Some of the axioms below will actually be schemata. A formula schema, or formula form, is a string G of the metalanguage that contains syntactic variables, such as A, P, f, a, t, x. Whenever we replace all these syntactic variables that occur in G by specific formulas, predicates, functions, constants, terms, or variables respectively, we obtain a specific well-formed formula, a so-called instance of the schema. For example, an instance of (∃x)x = a is (∃v12 )v12 = 0 (in the language of Peano arithmetic). An instance of A → A is v101 = v114 → v101 = v114 . I.3.13 Definition (Axioms and Axiom Schemata). The logical axioms are all the formulas in the group Ax1 and all the possible instances of the schemata in the remaining groups: Ax1. All formulas in Taut. Ax2. (Schema) A[x ← t] → (∃x)A

for any term t

By I.3.10–I.3.11, the notation already imposes a condition on t, that it is substitutable for x. N.B. We often see the above written as A[t] → (∃x)A[x] or even A[t] → (∃x)A Ax3. (Schema) For each object variable x, the formula x = x. Ax4. (Leibniz’s characterization of equality – first order version. Schema) For any formula A, object variable x, and terms t and s, the formula t = s → (A[x ← t] ↔ A[x ← s]) N.B. The above is written usually as t = s → (A[t] ↔ A[s]) We must remember that the notation already requires that t and s be free for x. We will denote the above set of logical axioms by . †




Plural of schema. This is of Greek origin, σ χ ηµα, ´ meaning – e.g., in geometry – figure or configuration or even formation.

I.3. Axioms and Rules of Inference

35

The logical axioms for equality are not the strongest possible, but they are adequate for the job. What Leibniz really proposed was the schema t = s ↔ (∀P)(P[t] ↔ P[s]), which says, intuitively, that “two objects t and s are equal iff, for every ‘property P’, both have P or neither has P”. Unfortunately, our system of notation (first-order language) does not allow quantification over predicate symbols (which can have as “values” arbitrary “properties”). But is not Ax4 read “for all formulas A” anyway? Yes, but with one qualification: “For all formulas A that we can write down in our system of notation”, and, alas, we cannot write all possible formulas of real mathematics down, because they are too many.† While the symbol “=” is suggestive of equality, it is not its shape that qualifies it as equality. It is the two axioms, Ax3 and Ax4, that make the symbol behave as we expect equality to behave, and any other symbol of any other shape (e.g., Enderton (1972) uses “≈”) satisfying these two axioms qualifies as formal equality that is intended to codify the metamathematical standard “=”. I.3.14 Remark. In Ax2 and Ax4 we imposed the condition that t (and s) must be substitutable in x. Here is why: Take A to stand for (∀y)x = y and B to stand for (∃y)¬x = y. Then, temporarily suspending the restriction on substitutability, A[x ← y] → (∃x)A is (∀y)y = y → (∃x)(∀y)x = y and x = y → (B ↔ B [x ← y]) is x = y → ((∃y)¬x = y ↔ (∃y)¬y = y) neither of which, obviously, is “valid”.‡ There is a remedy in the metamathematics: Move the quantified variable(s) out of harm’s way, by renaming them so that no quantified variable in A has the same name as any (free, of course) variable in t (or s). This renaming is formally correct (i.e., it does not change the meaning of the formula), as we will see in the variant (meta)theorem (I.4.13). Of course, †

‡

“Uncountably many”, in a precise technical sense developed in the chapter on cardinality in volume 2 (see p. 62, of this volume for a brief informal “course” in cardinality). This is due to Cantor’s theorem, which implies that there are uncountably many subsets of N. Each such subset A gives rise to the formula x ∈ A in the metalanguage. On the other hand, set theory’s formal system of notation, using just ∈ and U as start-up (nonlogical) symbols, is only rich enough to write down a countably infinite set of formulas (cf. p. 62). Thus, our notation will fail to denote uncountably many “real formulas” x ∈ A. Speaking intuitively is enough for now. Validity will be defined carefully pretty soon.

36

I. Basic Logic

it is always possible to effect this renaming, since we have countably many variables, and only finitely many appear free in t (and s) and A. This trivial remedy allows us to render the conditions in Ax2 and Ax4 harmless. Essentially,
a t (or s) is always substitutable after renaming. It is customary to assume a Platonist metatheory, and we do so. We can then say “countably many” variables without raising any eyebrows. Alternatively, we know how to get a new variable that is different from all those in a given finite set of variables without invoking an infinite supply. I.3.15 Definition (Rules of Inference). The following are the two rules of inference. These rules are relations in the sense of Section I.2, with inputs from the set Wff and outputs also in Wff. They are written traditionally as “fractions”. We call the “numerator” the premise(s) and the “denominator” the conclusion. We say that a rule of inference is applied to the formula(s) in the numerator, and that it yields (or results in) the formula in the denominator. Inf1. Modus ponens, or MP. For any formulas A and B , A, A → B B Inf2. ∃-introduction – pronounced E-introduction. For any formulas A and B such that x is not free in B , A→B (∃x)A → B N.B. Recall the conventions on eliminating brackets!




It is immediately clear that the definition above meets our requirement that the rules of inference be “algorithmic”, in the sense that whether they are applicable or how they are applicable can be decided and carried out in a finite number of steps by just looking at the form of (potential input) formulas (not at the “meaning” of such formulas). We next define -theorems, that is, formulas we can prove from the set of formulas  (this  may be empty). I.3.16 Definition (-Theorems). The set of -theorems, Thm , is the least inclusive subset of Wff that satisfies: Th1.  ⊆ Thm (cf. I.3.13). Th2.  ⊆ Thm . We call every member of  a nonlogical axiom. Th3. Thm is closed under each rule Inf1–Inf2.

I.3. Axioms and Rules of Inference

37

The metalinguistic statement A ∈ Thm is traditionally written as   A, and we say that A is proved from  or that it is a -theorem. We also say that A is deduced by , or that  deduces A. If  = ∅, then rather than ∅  A we write  A. We often say in this case that A is absolutely provable (or provable with no nonlogical axioms). We often write A, B , . . . , D  E for {A, B , . . . , D }  E .




I.3.17 Definition (-Proofs). We just saw that Thm is Cl(I , R), where I is the set of all logical and nonlogical axioms, and R contains just the two rules of inference. An (I , R)-derivation is also called a -proof (or just proof, if  is understood).
I.3.18 Remark. (1) It is clear that if each of A1 , . . . , An has a -proof and B has an {A1 , . . . , An }-proof, then B has a -proof. Indeed, simply concatenate all of the given -proofs (in any sequence). Append to the right of that sequence the given {A1 , . . . , An }-proof (that ends with B ). Then the entire sequence is a -proof, and ends with B . We refer to this phenomenon as the transitivity of . N.B. Transitivity of  allows one to invoke previously proved (by him or others) theorems in the course of a proof. Thus, practically, a -proof is a sequence of formulas in which each formula is an axiom, is a known -theorem, or is obtained by applying a rule of inference on previous formulas of the sequence. (2) If  ⊆  and   A, then also   A, as follows from I.3.16 or I.3.17. In particular,  A implies   A for any . (3) It is immediate from the definitions that for any formulas A and B , A, A → B  B

(i)

and if, moreover, x is not free in B , A → B  (∃x)A → B

(ii)

Some texts (e.g., Sch¨utte (1977)) give the rules in the format of (i)–(ii) above.


The axioms and rules provide us with a calculus, that is, a means to “calculate” proofs and theorems. In the interest of making the calculus more userfriendly – and thus more easily applicable to mathematical theories of interest, such as Peano arithmetic or set theory – we are going to develop in the next section a number of derived principles. These principles are largely of the form

38

I. Basic Logic

A1 , . . . , An  B . We call such a (provable in the metatheory) principle a derived rule of inference, since, by transitivity of , it can be used as a proof-step in a -proof. By contrast, the rules Inf1–Inf2 are “basic” or “primary”; they are given outright. We can now fix our understanding of the concept of a formal or mathematical theory. A (first order) formal (mathematical) theory over a language L, or just theory over L, or just theory, is a tuple (of “ingredients”) T = (L , , I, T ), where L is a first order language,  is a set of logical axioms, I is a set of rules of inference, and T a non-empty subset of Wff that is required to contain  (i.e.,  ⊆ T ) and be closed under the rules I. Equivalently, one may simply require that T is closed under , that is, for any  ⊆ T and any formula A, if   A, then A ∈ T . This is, furthermore, equivalent to requiring that A∈T

iff

T A

(1)

Indeed, the if direction follows from closure under , while the only if direction is a consequence of Definition I.3.16. T is the set of the formulas of the theory,† and we often say “a theory T ”, taking everything else for granted. If T = Wff, then the theory T is called inconsistent or contradictory. Otherwise it is called consistent. Throughout our exposition we fix  and I as in Definitions I.3.13 and I.3.15. By (1), T = ThmT . This observation suggests that we call theories such as the ones we have just defined axiomatic theories, in that a set  always exists so that T = Thm (if at a loss, we can just take  = T ). We are mostly interested in theories T for which there is a “small” set  (“small” by comparison with T ) such that T = Thm . We say that T is axiomatized by . Naturally, we call T the set of theorems, and  the set of nonlogical axioms of T. If, moreover,  is recognizable (i.e., we can tell “algorithmically” whether or not a formula A is in ), then we say that T is recursively axiomatized. Examples of recursively axiomatized theories are ZFC set theory and Peano arithmetic. On the other hand, if we take T to be all the formulas of arithmetic that are true when interpreted “in the intended way”‡ over N – the so-called † ‡

As opposed to “of the language”, which is all of Wff. That is, the symbol “0” of the language is interpreted as the 0 ∈ N, “Sx” as x + 1, “(∃x)” as “there is an x ∈ N”, etc.

I.3. Axioms and Rules of Inference

39

complete arithmetic – then there is no recognizable  such that T = Thm . We say that complete arithmetic is not recursively axiomatizable.† Pause. Why does complete arithmetic form a theory? Because work of Section I.5 – in particular, the soundness theorem – entails that it is closed under . We tend to further abuse language and call axiomatic theories by the name of their (set of) nonlogical axioms . Thus if T = (L , , I, T ) is a first order theory and T = Thm , then we may say interchangeably “theory T ”, “theory T ” or “theory ”. If  = ∅, then we have a pure or absolute theory (i.e., we are “just doing logic, not math”). If  = ∅, then we have an applied theory. Argot. A final note on language versus metalanguage, and theory versus metatheory. When are we speaking the metalanguage and when are we speaking the formal language? The answer is, respectively, “almost always” and “almost never”. As it has been remarked before, in principle, we are speaking the formal language exactly when we are pronouncing or writing down a string from Term or Wff. Otherwise we are (speaking or writing) in the metalanguage. It appears that we (and everybody else who has written a book in logic or set theory) is speaking and writing within the metalanguage with a frequency approaching 100%. The formalist is clever enough to simplify notation at all times. We will seldom be caught writing down a member of Wff in this book, and, on the rare occasions we may do so, it will only be to serve as an illustration of why one should avoid writing down such formulas: because they are too long and hard to read and understand. We will be speaking the formal language with a heavy “accent” and using many “idioms” borrowed from “real” (meta)mathematics and English. We will call our dialect argot, following Manin (1977). The important thing to remember is when we are working in the theory,‡ and this is precisely when we generate theorems. That is, it does not matter if a theorem (and much of the what we write down during the proof) is written in argot. Two examples: (1) One is working in formal number theory (or formal arithmetic) if one states and proves (say, from the Peano axioms) that “every natural number n > 1 † ‡

The trivial solution – that is, taking  = T – will not do, for it turns out that T is not recognizable. Important, because arguing in the theory restricts us to use only its axioms (and earlier proved theorems; cf. I.3.18) and its rules of inference – nothing extraneous to these syntactic tools is allowed.

40

I. Basic Logic has a prime factor”. Note how this theorem is stated in argot. Below we give its translation into the formal language of arithmetic:† (∀n)(S0 < n → (∃x)(∃y)(n = x × y ∧ S0 < x ∧ (∀m)(∀r )(x = m × r → m = S0 ∨ m = x)))

(1)

(2) One is working in formal logic if one is writing a proof of (∃v13 )v13 = v13 . Suppose though that our activity consists of effecting definitions, introducing axioms, or analyzing the behaviour or capability of T, e.g., proving some derived rule A1 , . . . , An  B – that is, a theorem schema – or investigating consistency‡ or “relative consistency”.§ Then we are operating in the metatheory, that is, in “real” mathematics. One of the most important problems posed in the metatheory is “Given a theory T and a formula A. Is A a theorem of T?” This is Hilbert’s Entscheidungsproblem, or decision problem. Hilbert believed that every recursively axiomatized theory ought to admit a “general” solution, by more or less mechanical means, to its decision problem. The techniques of G¨odel and the insight of Church showed that this problem is, in general, algorithmically unsolvable. As we have already stated (p. 36), metamathematics exists outside and independently of our effort to build this or that formal system. All its methods are – in principle – available to us for use in the analysis of the behaviour of a formal system. Pause. But how much of real mathematics are we allowed to use, reliably, to study or speak about the “simulator” that the formal system is?¶ For example, have we not overstepped our license by using induction (and, implicitly, the entire infinite set N) in our Platonist metatheory, specifically in the recursive or inductive definitions of terms, well-formed formulas, theorems, etc.? The quibble here is largely “political”. Some people argue (a major proponent of this was Hilbert) as follows: Formal mathematics was meant to crank out “true” statements of mathematics, but no “false” ones, and this freedom † ‡ § ¶

Well, almost. In the interest of brevity, all the variable names used in the displayed formula (1) are metasymbols. That is, whether or not T = Wff. That is, “if  is consistent” – where we are naming the theory by its nonlogical axioms – “does it stay so after we have added some formula A as a nonlogical axiom?” The methods or scope of the metamathematics that a logician uses – in the investigation of some formal system – are often restricted for technical or philosophical reasons.

I.3. Axioms and Rules of Inference

41

from contradiction ought to be verifiable. Now, as we are so verifying in the metatheory (i.e., outside the formal system) shouldn’t the metatheory itself be “above suspicion” (of contradiction, that is)? Naturally. Hilbert’s suggestion for achieving this “above suspicion” status was, essentially, to utilize in the metatheory only a small fragment of “reality” that is so simple and close to intuition that it does not need itself a “certificate” (via formalization) for its freedom from contradiction. In other words, restrict the metamathematics.† Such a fragment of the metatheory, he said, should have nothing to do with the infinite, in particular with the entire set N and all that it entails (e.g., inductive definitions and proofs).‡ If it were not for G¨odel’s incompleteness results, this position – that metamathematical techniques must be finitary – might have prevailed. However, G¨odel proved it to be futile, and most mathematicians have learnt to feel comfortable with infinitary metamathematical techniques, or at least with N and induction.§ Of course, it would be reckless to use as metamathematical tools “mathematics” of suspect consistency (e.g., the full na¨ıve theory of sets). It is worth pointing out that one could fit (with some effort) our inductive definitions within Hilbert’s style. But we will not do so. First, one would have to abandon the elegant (and now widely used) approach with closures, and use instead the concept of derivations of Section I.2. Then one would somehow have to effect and study derivations without the benefit of the entire set N. Bourbaki (1966b, p. 15) does so with his constructions formatives. Hermes (1973) is another author who does so, with his “term-” and “formula-calculi” (such calculi being, essentially, finite descriptions of derivations). Bourbaki (but not Hermes) avoids induction over all of N. In his metamathematical discussions of terms and formulas¶ that are derived by a derivation †

‡

§

¶

Otherwise we would need to formalize the metamathematics – in order to “certify” it – and next the metametamathematics, and so on. For if “metaM” is to authoritatively check “M” for consistency, then it too must be consistent; so let us formalize “metaM” and let “metametaM” check it; . . . a never ending story. See Hilbert and Bernays (1968, pp. 21–29) for an elaborate scheme that constructs “concrete number objects” – Ziffern or “numerals” – “|”, “||”, “|||”, etc., that stand for “1”, “2”, “3”, etc., complete with a “concrete mathematical induction” proof technique on these objects, and even the beginnings of their recursion theory. Of course, at any point, only finite sets of such objects were considered. Some proponents of infinitary techniques in metamathematics have used very strong words in describing the failure of “Hilbert’s program”. Rasiowa and Sikorski (1963) write in their introduction: “However G¨odel’s results exposed the fiasco of Hilbert’s finitistic methods as far as consistency is concerned.” For example, in loc. cit., p. 18, where he proves that, in our notation, A[x ← y] and t[x ← y] are a formula and term respectively.

42

I. Basic Logic

d1 , . . . , dn , he restricts his induction arguments on the segment {0, 1, . . . , n}, that is, he takes an I.H. on k < n and proceeds to k + 1.

I.4. Basic Metatheorems We are dealing with an arbitrary theory T = (L , , I, T ), such that  is the set of logical axioms (I.3.13) and I are the inference rules (I.3.15). We also let  be an appropriate set of nonlogical axioms, i.e., T = Thm . I.4.1 Metatheorem (Post’s “Extended” Tautology Theorem). If A1 , . . . ,
An |=Taut B then A1 , . . . , An  B . Proof. The assumption yields that |=Taut A1 → · · · → An → B

(1)

Thus, since the formula in (1) is in , using Definition I.3.16, we have A1 , · · · , An  A1 → · · · → An → B Applying modus ponens to (2) n times, we deduce B .

(2)


I.4.1 is an omnipresent derived rule. I.4.2 Definition. A and B provably equivalent in T means that   A ↔ B . I.4.3 Metatheorem. Any two theorems A and B of T are provably equivalent in T. Proof. By I.4.1,   A yields   B → A. Similarly,   B yields   A → B . One more application of I.4.1 yields   A ↔ B .
Worth noting:  ¬x = x ↔ ¬y = y (why?), but neither ¬x = x nor ¬y = y is a ∅-theorem. I.4.4 Remark (Hilbert Style Proofs). In practice we write proofs “vertically”, that is, as numbered vertical sequences (or lists) of formulas. The numbering helps the annotational comments that we insert to the right of each formula that we list, as the following proof demonstrates.

I.4. Basic Metatheorems

43

A metatheorem admits a metaproof, strictly speaking. The following is a derived rule (or theorem schema) and thus belongs to the metatheory (and so does its proof). Another point of view is possible, however: The syntactic symbols x, A, and B below stand for a specific variable and specific formulas that we just forgot to write down explicitly. Then one can think of the proof as a (formal)
Hilbert style proof. I.4.5 Metatheorem (∀-Introduction – Pronounced “A-Introduction”). If x does not occur free in A, then A → B  A → (∀x)B . Proof. (1) (2) (3) (4) (5)

A→B ¬B → ¬A (∃x)¬B → ¬A A → ¬(∃x)¬B A → (∀x)B

given (1) and I.4.1 (2) and ∃-introduction (3) and I.4.1 (4), introducing the ∀-abbreviation




I.4.6 Metatheorem (Specialization). For any formula A and term t,  (∀x)A → A [t]. At this point, the reader may want to review our abbreviation conventions, in particular, see Ax2 (I.3.13). Proof. (1) (2) (3)

¬A[t] → (∃x)¬A ¬(∃x)¬A → A[t] (∀x)A → A [t]

in  (1) and I.4.1 (2), introducing the ∀-abbreviation




I.4.7 Corollary. For any formula A,  (∀x)A → A . Proof. A[x ← x] ≡ A.




Pause. Why is A[x ← x] the same string as A? I.4.8 Metatheorem (Generalization). For any  and any A, if   A, then   (∀x)A .

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I. Basic Logic

Proof. Choose y ≡ x. Then we continue any given proof of A (from ) as follows: (1) (2) (3) (4) (5)

A y=y→A y = y → (∀x)A y=y (∀x)A

proved from  (1) and I.4.1 (2) and ∀-introduction in  (3), (4), and MP




I.4.9 Corollary. For any  and any A,   A iff   (∀x)A. Proof. By I.4.7, I.4.8, and modus ponens.




I.4.10 Corollary. For any A, A  (∀x)A and (∀x)A  A.




The above corollary motivates the following definition. It also justifies the common mathematical practice of the “implied universal quantifier”. That is, we often state “. . . x . . . ” when we mean “(∀x) . . . x . . . ”. I.4.11 Definition (Universal Closure). Let y1 , . . . , yn be the list of all free variables of A. The universal closure of A is the formula (∀y1 )(∀y2 ) · · · (∀yn )A –
often written more simply as (∀y1 y2 . . . yn )A or even (∀yn )A. By I.4.10, a formula deduces and is deduced by its universal closure. Pause. We said the universal closure. Hopefully, the remark immediately above is undisturbed by permutation of (∀y1 )(∀y2 ) · · · (∀yn ). Is it? (Exercise I.11). I.4.12 Corollary (Substitution of Terms). A[x1 , . . . , xn ]  A[t1 , . . . , tn ] for any terms t1 , . . . , tn . The reader may wish to review I.3.12 and the remark following it. Proof. We illustrate the proof for n = 2. What makes it interesting is the requirement to have “simultaneous substitution”. To that end we first substitute into x1 and x2 new variables z, w – i.e., not occurring in either A or in the ti . The proof is the following sequence. Comments justify, in each case, the presence of the formula immediately to the left by virtue of the presence of the

I.4. Basic Metatheorems

45

immediately preceding formula. A[x1 , x2 ] (∀x1 )A[x1 , x2 ] A[z, x2 ] (∀x2 )A[z, x2 ] A[z, w]

starting point generalization specialization; x1 ← z generalization specialization; x2 ← w

Now z ← t1 , w ← t2 , in any order, is the same as “simultaneous substitution I.3.12”: (∀z)A[z, w] A[t1 , w] (∀w)A[t1 , w] A[t1 , t2 ]

generalization specialization; z ← t1 generalization specialization; w ← t2




I.4.13 Metatheorem (The Variant, or Dummy-Renaming, Metatheorem). For any formula (∃x)A, if z does not occur in it (i.e., is neither free nor bound), then  (∃x)A ↔ (∃z)A[x ← z]. We often write this (under the stated conditions) as  (∃x)A[x] ↔ (∃z)A[z]. By the way, another way to state the conditions is “if z does not occur in A (i.e., is neither free nor bound in A), and is different from x”. Of course, if z ≡ x, then there is nothing to prove. Proof. Since z is substitutable in x under the stated conditions, A[x ← z] is defined. Thus, by Ax2,  A[x ← z] → (∃x)A By ∃-introduction – since z is not free in (∃x)A – we also have  (∃z)A[x ← z] → (∃x)A

(1)

We note that x is not free in (∃z)A[x← z] and is free for z in A[x← z]. Indeed, A[x ← z][z ← x] ≡ A. Thus, by Ax2,  A → (∃z)A[x ← z] Hence, by ∃-introduction,  (∃x)A → (∃z)A[x ← z] Tautological implication from (1) and (2) concludes the argument.

(2)


46

I. Basic Logic

Why is A[x ← z][z ← x] ≡ A? We can see this by induction on A (recall that z occurs as neither free nor bound in A). If A is atomic, then the claim is trivial. The claim also clearly “propagates” with the propositional formation rules, that is, I.1.8(b). Consider then the case that A ≡ (∃w)B . Note that w ≡ x is possible under our assumptions, but w ≡ z is not. If w ≡ x, then A[x ← z] ≡ A; in particular, z is not free in A; hence A[x ← z][z ← x] ≡ A as well. So let us work with w ≡ x. By I.H., B [x ← z][z ← x] ≡ B . Now A[x ← z][z ← x] ≡ ((∃w)B )[x ← z][z ← x] ≡ ((∃w)B [x ← z])[z ← x] ≡ ((∃w)B [x ← z][z ← x]) ≡ ((∃w)B ) ≡A

see I.3.11; w ≡ z see I.3.11; w ≡ x I.H.

By I.4.13, the issue of substitutability becomes moot. Since we have an infinite supply of variables (to use, for example, as bound variables), we can always change the names of all the bound variables in A so that the new names are different from all the free variables in A or t. In so doing we obtain a formula B that is (absolutely) provably equivalent to the original. Then B [x ← t] will be defined (t will be substitutable in x). Thus, the moral is: any term t is free for x in A after an appropriate ‘dummy’ renaming. By the way, this is one of the reasons we want an infinite supply (or an extendible finite set, for the finitist) of formal variables. I.4.14 Definition. In the following we will often discuss two (or more) theories at once. Let T = (L , , I, T ) and T = (L , , I, T ) be two theories, such that V ⊆ V . This enables T to be “aware” of all the formulas of T (but not vice versa, since L contains additional nonlogical symbols). We say that T is an extension of T (in symbols, T ≤ T ) iff T ⊆ T . Let A be a formula over L (so that both theories are aware of it). The symbols T A and T A are synonymous with A ∈ T and A ∈ T respectively. Note that we did not explicitly mention the nonlogical axioms  or  to the left of , since the subscript of  takes care of that information. We say that the extension is conservative iff for any A over L, whenever T A it is also the case that T A. That is, when it comes to formulas over the language (L) that both theories understand, then the new theory does not
do any better than the old in producing theorems. I.4.15 Metatheorem (Metatheorem on Constants). Let us extend a language L of a theory T by adding new constant symbols e1 , . . . , en to the alphabet V , resulting in the alphabet V , language L , and theory T .

I.4. Basic Metatheorems

47

Furthermore, assume that  = , that is, we did not add any new nonlogical axioms. Then T A[e1 , . . . , en ] implies T A[x1 , . . . , xn ] for any variables x1 , . . . , xn that occur nowhere in A[e1 , . . . , en ], as either free or bound variables. Proof. Fix a set of variables x1 , . . . , xn as described above. We do induction on T -theorems. Basis. A[e1 , . . . , en ] is a logical axiom (over L ); hence so is A[x1 , . . . , xn ], over L – because of the restriction on the xi – thus T A[x1 , . . . , xn ]. Note that A[e1 , . . . , en ] cannot be nonlogical under our assumptions. Pause. What does the restriction on the xi have to do with the claim above? Modus ponens. Here T B [e1 , . . . , en ] → A[e1 , . . . , en ] and T B [e1 , . . . , en ]. By I.H., T B [y1 , . . . , yn ] → A[y1 , . . . , yn ] and T B [y1 , . . . , yn ], where y1 , . . . , yn occur nowhere in B [e1 , . . . , en ] → A[e1 , . . . , en ] as either free or bound variables. By modus ponens, T A[y1 , . . . , yn ]; hence T A[x1 , . . . , xn ] by I.4.12 (and I.4.13). ∃-introduction. We have T B [e1 , . . . , en ] → C [e1 , . . . , en ], z is not free in C [e1 , . . . , en ], and A[e1 , . . . , en ] ≡ (∃z)B [e1 , . . . , en ] → C [e1 , . . . , en ]. By I.H., if w1 , . . . , wn – distinct from z – occur nowhere in B [e1 , . . . , en ] → C [e1 , . . . , en ] as either free or bound, then we get T B [w1 , . . . , wn ] → C [w1 , . . . , wn ]. By ∃-introduction we get T (∃z)B [w1 , . . . , wn ] → C [w1 , . . . , wn ]. By I.4.12 and I.4.13 we get T (∃z)B [x1 , . . . , xn ] → C [x1 , . . . , xn ], i.e., T A[x1 , . . . , xn ].
I.4.16 Corollary. Let us extend a language L of a theory T by adding new constant symbols e1 , . . . , en to the alphabet V , resulting to the alphabet V , language L , and theory T . Furthermore, assume that  = , that is, we did not add any new nonlogical axioms. Then T A[e1 , . . . , en ] iff T A[x1 , . . . , xn ], for any choice of variables
x1 , . . . , xn . Proof. If part: Trivially, T A[x1 , . . . , xn ] implies T A[x1 , . . . , xn ], hence T A[e1 , . . . , en ] by I.4.12. Only-if part: Choose variables y1 , . . . , yn that occur nowhere in A[e1 , . . . , en ] as either free or bound. By I.4.15, T A[y1 , . . . , yn ]; hence,
by I.4.12 and I.4.13, T A[x1 , . . . , xn ]. I.4.17 Remark. Thus, the extension T of T is conservative, for, if A is over L, then A[e1 , . . . , en ] ≡ A. Therefore, if T A, then T A[e1 , . . . , en ]; hence T A[x1 , . . . , xn ], that is, T A.

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A more emphatic way to put the above is this: T is not aware of any new nonlogical facts that T did not already “know” although by a different name. If T can prove A[e1 , . . . , en ], then T can prove the same “statement”, however, using (any) names (other than the ei ) that are meaningful in its own language; namely, it can prove A[x1 , . . . , xn ].
The following corollary stems from the proof (rather than the statement) of I.4.15 and I.4.16, and is important. I.4.18 Corollary. Let e1 , . . . , en be constants that do not appear in the nonlogical axioms . Then, if x1 , . . . , xn are any variables, and if   A[e1 , . . . , en ], it is also the case that   A[x1 , . . . , xn ]. I.4.19 Metatheorem (The Deduction Theorem). For any closed formula A, arbitrary formula B , and set of formulas , if +A  B , then   A → B . N.B.  + A denotes the augmentation of  by adding the formula A. In the present metatheorem A is a single (but unspecified) formula. However, the notation extends to the case where A is a schema, in which case it means the augmentation of  by adding all the instances of the schema. A converse of the metatheorem is also true trivially: That is,   A → B implies  + A  B . This direction immediately follows by modus ponens and does not require the restriction on A. Proof. The proof is by induction on  + A theorems. Basis. Let B be logical or nonlogical (but, in the latter case, assume B ≡ A). Then   B . Since B |=Taut A → B , it follows by I.4.1 that   A → B . Now, if B ≡ A, then A → B is a logical axiom (group Ax1); hence   A → B once more. Modus ponens. Let  + A  C , and  + A  C → B . By I.H.,   A → C and   A → C → B . Since A → C , A → C → B |=Taut A → B , we have   A → B . ∃-introduction. Let  + A  C → D , and B ≡ (∃x)C → D , where x is not free in D . By I.H.,   A → C → D . By I.4.1,   C → A → D ; hence   (∃x)C → A → D by ∃-introduction (A is closed). One more application of I.4.1 yields   A → (∃x)C → D .
I.4.20 Remark. (1) Is the restriction that, A must be closed important? Yes. Let A ≡ x = a, where “a” is some constant. Then, even though A  (∀x)A

I.4. Basic Metatheorems

49

by generalization, it is not always true that  A → (∀x)A. This follows from soundness considerations (next section). Intuitively, assuming that our logic “doesn’t lie” (that is, it proves no “invalid” formulas), we immediately infer that x = a → (∀x)x = a cannot be absolutely provable, for it is a “lie”. It fails at least over N, if a is interpreted to be “0”. (2) I.4.16 adds flexibility to applications of the deduction theorem: T (A → B )[x1 , . . . , xn ]

(∗)

where [x1 , . . . , xn ] is the list of all free variables just in A, is equivalent (by I.4.16) to T (A → B )[e1 , . . . , en ]

(∗∗)

where e1 , . . . , en are new constants added to V (with no effect on nonlogical axioms:  =  ). Now, since A[e1 , . . . , en ] is closed, proving  + A[e1 , . . . , en ]  B [e1 , . . . , en ] establishes (∗∗), and hence also (∗). In practice, one does not perform this step explicitly, but ensures that, throughout the  + A proof, whatever free variables were present in A “behaved like constants”, or, as we also say, were frozen. (3) In some expositions the deduction theorem is not constrained by requiring that A be closed (e.g., Bourbaki (1966b) and more recently Enderton (1972)). Which version is right? Both are in their respective contexts. If all the rules of inference are “propositional” (e.g., as in Bourbaki (1966b) and Enderton (1972), who only employ modus ponens) – that is, they do not meddle with quantifiers – then the deduction theorem is unconstrained. If, on the other hand, the rules of inference manipulate object variables via quantification, then one cannot avoid constraining the application of the deduction theorem, lest one want to derive (the invalid)  A → (∀x)A from the valid A  (∀x)A. This also entails that approaches such as in Bourbaki (1966b) and Enderton (1972) do not allow “full” generalization “A  (∀x)A ”. They only allow a “weaker” rule, “if  A, then  (∀x)A”.† (4) This divergence of approach in choosing rules of inference has some additional repercussions: One has to be careful in defining the semantic counterpart

†

Indeed, they allow a bit more generally, namely, the rule “if   A with a side condition, then   (∀x)A. The side condition is that the formulas of  do not have free occurrences of x.” Of course,  can be always taken to be finite (why?), so that this condition is not unrealistic.

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of , namely, |= (see next section). One wants the two symbols to “track each other” faithfully (G¨odel’s completeness theorem).† I.4.21 Corollary (Proof by Contradiction). Let A be closed. Then   A iff  + ¬A is inconsistent. Proof. If part: Given that T = Wff, where T is the theory  + ¬A. In particular,  + ¬A  A. By the deduction theorem,   ¬A → A. But ¬A → A |=Taut A. Only-if part: Given that   A. Hence  + ¬A  A as well (recall I.3.18(2)). Of course,  + ¬A  ¬A. Since A, ¬A |=Taut B for an
arbitrary B , we are done. Pause. Is it necessary to assume that A is closed in I.4.21? Why? The following is important enough to merit stating. It follows from the type of argument we employed in the only-if part above. I.4.22 Metatheorem. T is inconsistent iff for some A, both T A and T ¬A hold. We also list below a number of “quotable” proof techniques. These techniques are routinely used by mathematicians, and will be routinely used by us. The proofs of all the following metatheorems are delegated to the reader. I.4.23 Metatheorem (Distributivity or Monotonicity of ∃). For any x, A, B , A → B  (∃x)A → (∃x)B Proof. See Exercise I.12.




I.4.24 Metatheorem (Distributivity or Monotonicity of ∀). For any x, A, B , A → B  (∀x)A → (∀x)B Proof. See Exercise I.13.




The term “monotonicity” is inspired by thinking of “→” as “≤”. How? Well, we have the tautology (A → B ) ↔ (A ∨ B ↔ B) †

In Mendelson (1987), |= is defined inconsistently with .

(i)

I.4. Basic Metatheorems

51

If we think of “A ∨ B ” as “max(A, B )”, then the right hand side in (i) above says that B is the maximum of A and B , or that A is “less than or equal to” B . The above metatheorems say that both ∃ and ∀ preserve this “inequality”. I.4.25 Metatheorem (Equivalence Theorem, or Leibniz Rule). Let   A ↔ B , and let C be obtained from C by replacing some – possibly, but not necessarily, all – occurrences of a subformula A of C by B . Then   C ↔ C , i.e., A↔B C ↔C is a derived rule. Proof. The proof is by induction on formulas C . See Exercise I.15.




Equational or calculational predicate logic is a particular foundation of first order logic that uses the above Leibniz rule as the primary rule of inference. In “practising” such logic one prefers to write proofs as chains of equivalences. Most equivalences in such a chain stem from an application of the rule. See Dijkstra and Scholten (1990), Gries and Schneider (1994), Tourlakis (2000a, 2000b, 2001b). I.4.26 Metatheorem (Proof by Cases). Suppose that   A1 ∨ · · · ∨ An , and   Ai → B for i = 1, . . . . Then   B . Proof. Immediate, by I.4.1.




Proof by cases usually benefits from the application of the deduction theorem. That is, having established   A1 ∨ · · · ∨ An , one then proceeds to adopt, in turn, each Ai (i = 1, . . . , n) as a new nonlogical axiom (with its variables “frozen”). In each “case” (Ai ) one proceeds to prove B . At the end of all this one has established   B . In practice we normally use the following argot: “We will consider cases Ai , for i = 1, . . . , n.† Case A1 .

. . . therefore, B .‡

··· Case An . † ‡

. . . therefore, B .”

To legitimize this splitting into cases, we must, of course, show   A1 ∨ · · · ∨ An . That is, we add the axiom A1 to , freezing its variables, and we then prove B .

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I.4.27 Metatheorem (Proof by Auxiliary Constant). Suppose that for arbitrary A and B over the language L we know (1)   (∃x)A[x] (2)  + A[a]  B , where a is a new constant not in the language L of . Furthermore assume that in the proof of B all the free variables of A[a] were frozen. Then   B . Proof. Exercise I.21.




The technique that flows from this metatheorem is used often in practice. For example, in projective geometry axiomatized as in Veblen and Young (1916), in order to prove Desargues’s theorem on perspective triangles on the plane, we use some arbitrary point (this is the auxiliary constant!) off the plane, having verified that the axioms guarantee that such a point exists. It is important to note that Desargues’s theorem does not refer to this point at all – hence the term “auxiliary”. In this example, from projective geometry, “B ” is Desargues’s theorem, “(∃x)A[x]” asserts that there are points outside the plane, a is an arbitrary such point, and the proof (2) starts with words like “Let a be a point off the plane” – which is argot for “add the axiom A[a]”. I.5. Semantics; Soundness, Completeness, Compactness So what do all these symbols mean? We show in this section how to “decode” the formal statements (formulas) into informal statements of “real” mathematics. Conversely, this will entail an understanding of how to code statements of real mathematics in our formal language. The rigorous† definition of semantics for first order languages is due to Tarski and is often referred to as “Tarski semantics”. The flavour of the particular definition given below is that of Shoenfield (1967), and it accurately reflects our syntactic choices – most importantly, the choice to allow full generalization A  (∀x)A. In particular, we will define the semantic counterpart of , namely, |=, pronounced “logically implies”, to ensure that   A iff  |= A. This is the content of G¨odel’s completeness theorem, which we prove in this section. This section will place some additional demands on the reader’s recollection of notation and facts from informal set theory. We will, among other things, †

One often says “The formal definition of semantics . . . ”, but the word “formal” is misleading here, for we are actually defining semantics in the metatheory, not in the formal theory.

I.5. Semantics; Soundness, Completeness, Compactness

53

make use of notation from na¨ıve set theory, such as An

(or
A × · · · × A) n times

for the set of ordered n-tuples of members of A.

We will also use the symbols ⊆, ∪, a∈I .† In some passages – delimited by warning signs – these demands will border on the unreasonable. For example, in the proof of the G¨odel-Mal cev completeness-compactness result we will need some elementary understanding of ordinals – used as indexing tools – and cardinality. Some readers may not have such background. This prerequisite material can be attained by consulting a set theory book (e.g., the second volume of these lectures). I.5.1 Definition. Given a language L = (V , Term, Wff), a structure M = (M, I ) appropriate for L is such that M = ∅ is a set (the domain or underlying set or universe‡ ) and I (“I ” for interpretation) is a mapping that assigns (1) to each constant a of V a unique member a I ∈ M, (2) to each function f of V – of arity n – a unique (total)§ function f I : M n → M,
(3) to each predicate P of V – of arity n – a unique set P I ⊆ M n .¶ I.5.2 Remark. The structure M is often given more verbosely, in conformity with practice in algebra. Namely, one “unpacks” the I into a list a I , bI , . . . ; f I , g I , . . . ; P I , Q I , . . . and writes instead M = (M; a I , bI , . . . ; f I , g I , . . . ; P I , Q I , . . . ). Under this understanding, a structure is an underlying set (universe), M, along with a list of “concrete” constants, functions, and relations that “interpret” corresponding “abstract” items of the language. Under the latter notational circumstances we often use the symbols a M, f M, P – rather than a I , f I , P I – to indicate the interpretations in M of the constant a, function f , and predicate P respectively. M

†

‡ § ¶

If we have a set of sets {S

a , Sb , Sc , . . . }, where the indices a, b, c, . . . all come out of an “index all those objects

x that are found in set” I , then the symbol i∈I Si stands for the collection of

∞ Si instead of i∈N Si . A ∪ B is at least one

of the sets Si . It is a common habit to write i=0 the same as i∈{1,2} Si , where we have let S1 = A and S2 = B. Often the qualification “of discourse” is added to the terms “domain” and “universe”. Requiring f I to be total is a traditional convention. By the way, total means that f I is defined everywhere on M n . Thus P I is an n-ary relation with inputs and outputs in M.

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We have said above “structure appropriate for L”, thus emphasizing the generality of the language and therefore our ability to interpret what we say in it in many different ways. Often though, e.g., as in formal arithmetic or set theory, we have a structure in mind to begin with, and then build a formal language to formally codify statements about the objects in the structure. Under these circumstances, in effect, we define a language appropriate for the structure. We use the symbol L M to indicate that the language was built to fit the structure M.
I.5.3 Definition. We routinely add new nonlogical symbols to a language L to obtain a language L . We say that L is an extension of L and that L is a restriction of L . Suppose that M = (M, I ) is a structure for L, and let M = (M, I ) be a structure with the same underlying set M, but with I extended to I so that the latter gives meaning to all new symbols while it gives the same meaning, as I does, to the symbols of L. We call M an expansion (rather than “extension”) of M, and M a reduct (rather than “restriction”) of M . We may (often) write I = I  L to indicate that the “mapping” I – restricted to L (symbol “  ”) – equals I . We may also write M = M  L instead.
I.5.4 Definition. Given L and a structure M = (M, I ) appropriate for L. L(M) denotes the language obtained from L by adding to V a unique new name i for each object i ∈ M. This amends both sets Term, Wff into Term(M), Wff(M). Members of the latter sets are called M-terms and M-formulas respectively. I

We extend the mapping I to the new constants by: i = i for all i ∈ M
(where the “=” here is metamathematical: equality on M). All we have done here is to allow ourselves to do substitutions like [x ← i] formally. We do, instead, [x ← i]. One next gives “meaning” to all closed terms in L(M). The following uses definition by recursion (I.2.13) and relies on the fact that the rules that define terms are unambiguous. I.5.5 Definition. For closed terms t in Term(M) we define the symbol t I ∈ M inductively: (1) If t is any of a (original constant) or i (imported constant), then t I has already been defined. (2) If t is the string f t1 . . . tn , where f is n-ary, and t1 , . . . , tn are closed MI I
terms, we define t I to be the object (of M) f I (t1 , . . . , tn ). Finally, we give meaning to all closed M-formulas, again by recursion (over Wff).

I.5. Semantics; Soundness, Completeness, Compactness

55

I.5.6 Definition. For any closed formula A in Wff(M) we define the symbol AI inductively. In all cases, AI ∈ {t, f}. (1) If A ≡ t = s, where t and s are closed M-terms, then AI = t iff t I = s I . (The last two occurrences of “=” are metamathematical.) (2) If A ≡ Pt1 . . . tn , where P is an n-ary predicate and the ti are closed I I I I M-terms, then AI = t iff t1 , . . . , tn  ∈ P I or P I (t1 , . . . , tn ) “holds”. (Or “is true”; see p. 19. Of course, the last occurrence of “=” is metamathematical.) (3) If A is any of the sentences ¬B , B ∨ C , then AI is determined by the usual truth tables (see p. 30) using the values B I and C I . That is, (¬B )I = F¬ (B I ) and (B ∨ C )I = F∨ (B I , C I ). (The last two occurrences of “=” are metamathematical.) (4) If A ≡ (∃x)B , then AI = t iff (B [x ← i])I = t for some i ∈ M. (The
last two occurrences of “=” are metamathematical.) We have “imported” constants from M into L in order to be able to state the semantics of (∃x)B above in the simple manner we just did (following Shoenfield (1967)). We often state the semantics of (∃x)B by writing ((∃x)B [x])I is true

iff

(∃i ∈ M)(B [i])I is true

I.5.7 Definition. Let A ∈ Wff, and M be a structure as above. An M-instance of A is an M-sentence A(i 1 , . . . , i k ) (that is, all the free variables of A have been replaced by imported constants). We say that A is valid in M, or that M is a model of A, iff for all M-instances A of A it is the case that A I = t.† Under these circumstances we write |=M A. For any set of formulas  from Wff, the expression |=M , pronounced “M is a model of ”, means that for all A ∈ , |=M A. A formula A is universally valid or logically valid (we often say just valid ) iff every structure appropriate for the language is a model of A. Under these circumstances we simply write |= A. If  is a set of formulas, then we say it is satisfiable iff it has a model. It is finitely satisfiable iff every finite subset of  has a model.‡
The definition of validity of A in a structure M corresponds with the normal mathematical practice. It says that a formula is true (in a given “context” M) just in case it is so for all possible values of the free variables. † ‡

We henceforth discontinue our pedantic “(The last occurrence of “=” is metamathematical.)”. These two concepts are often defined just for sentences.

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I.5.8 Definition. We say that  logically implies A, in symbols  |= A, meaning that every model of  is also a model of A.
I.5.9 Definition (Soundness). A theory (identified by its nonlogical axioms)  is sound iff, for all A ∈ Wff,   A implies  |= A, that is, iff all the
theorems of the theory are logically implied by the nonlogical axioms. Clearly then, a pure theory T is sound iff T A implies |= A for all A ∈ Wff. That is, all its theorems are universally valid. Towards the soundness result† below we look at two tedious (but easy) lemmata. I.5.10 Lemma. Given a term t, variables x ≡ y, where y does not occur in t, and a constant a. Then, for any term s and formula A, s[x ← t][y ← a] ≡ s[y ← a][x ← t] and A[x ← t][y ← a] ≡ A[y ← a][x ← t]. Proof. Induction on s: Basis:

 if s    if s s[x ← t][y ← a] ≡ if s    if s

≡x ≡y ≡z ≡b

then t then a where x ≡ z ≡ y, then z then b

≡ s[y ← a][x ← t] For the induction step let s ≡ f r1 . . . rn , where f has arity n. Then s[x ← t][y ← a] ≡ f r1 [x ← t][y ← a] . . . rn [x ← t][y ← a] ≡ f r1 [y ← a][x ← t] . . . rn [y ← a][x ← t] ≡ s[y ← a][x ← t] Induction on A: Basis: A[x ← t][y ← a]   if A         ≡ if A         

≡ Pr1 . . . rn then Pr1 [x ← t][y ← a] . . . rn [x ← t][y ← a] ≡ Pr1 [y ← a][x ← t] . . . rn [y ← a][x ← t] ≡ r = s then r [x ← t][y ← a] = s[x ← t][y ← a] ≡ r [y ← a][x ← t] = s[y ← a][x ← t]

≡ A[y ← a][x ← t] †

by I.H.

Also nicknamed “the easy half of G¨odel’s completeness theorem”.

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57

The property we are proving, trivially, propagates with Boolean connectives. Let us do the induction step just in the case where A ≡ (∃w)B . If w ≡ x or w ≡ y, then the result is trivial. Otherwise, A[x ← t][y ← a] ≡ ((∃w)B )[x ← t][y ← a] ≡ ((∃w)B [x ← t][y ← a]) ≡ ((∃w)B [y ← a][x ← t])

by I.H.

≡ ((∃w)B )[y ← a][x ← t] ≡ A[y ← a][x ← t]




I.5.11 Lemma. Given a structure M = (M, I ), a term s, and a formula A, both over L(M). Suppose each of s and A have at most one free variable, x. Let t be a closed term over L(M) such that t I = i ∈ M. Then (s[x ← I t]) = (s[x ← i])I and (A[x ← t])I = (A[x ← i])I . Of course, since t is closed, A[x ← t] is defined. Proof. Induction on s: Basis. s[x ← t] ≡ s if s ∈ {y, a, j} (y ≡ x). Hence (s[x ← t])I = s I = (s[x ← i])I in this case. If s ≡ x, then s[x ← t] ≡ t and s[x ← i] ≡ i, and the claim follows once more. For the induction step let s ≡ f r1 . . . rn , where f has arity n. Then (s[x ← t])I = f I ((r1 [x ← t])I , . . . , (rn [x ← t])I ) = f I ((r1 [x ← i])I , . . . , (rn [x ← i])I ) = (s[x ← i])I

by I.H.

Induction on A: Basis. If A ≡ Pr1 . . . rn , then† (A[x ← t])I = P I ((r1 [x ← t])I , . . . , (rn [x ← t])I ) = P I ((r1 [x ← i])I , . . . , (rn [x ← i])I ) = (A[x ← i])I Similarly if A ≡ r = s. The property we are proving, clearly, propagates with Boolean connectives. Let us do the induction step just in the case where A = (∃w)B . If w ≡ x, the result is trivial. Otherwise, we note that – since t is closed – w does not occur †

For a metamathematical relation Q, as is usual (p. 19), Q(a, b, . . . ) = t, or just Q(a, b, . . . ), stands for a, b, . . .  ∈ Q.

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in t, and proceed as follows: (A[x ← t])I = t iff iff iff iff iff iff iff iff iff iff iff

(((∃w)B )[x ← t])I = t (((∃w)B [x ← t]))I = t (B [x ← t][w ← j])I = t for some j (B [w ← j][x ← t])I = t for some j ((B [w ← j])[x ← t])I = t for some ((B [w ← j])[x ← i])I = t for some (B [w ← j][x ← i])I = t for some j (B [x ← i][w ← j])I = t for some j (((∃w)B [x ← i]))I = t by I.5.6(4) (((∃w)B )[x ← i])I = t (A[x ← i])I = t

∈ M, by I.5.6(4) ∈ M, by I.5.10 j∈M j ∈ M, by I.H. ∈M ∈ M, by I.5.10


I.5.12 Metatheorem (Soundness). Any first order theory (identified by its nonlogical axioms) , over some language L, is sound. Proof. By induction on -theorems, A, we prove that  |= A. That is, we fix a structure for L, say M, and assume that |=M . We then proceed to show that |=M A. Basis. A is a nonlogical axiom. Then our conclusion is part of the assumption, by I.5.7. Or A is a logical axiom. There are a number of cases: Case 1. |=Taut A. We fix an M-instance of A, say A , and show that A I = t. Let p1 , . . . , pn be all the propositional variables (alias prime I formulas) occurring in A . Define a valuation v by setting v( pi ) = pi I for i = 1, . . . , n. Clearly, t = v¯ (A ) = A (the first “=” because |=Taut A , the second because after prime formulas were taken care of, all that remains to be done for the evaluation of A I is to apply Boolean connectives – see I.5.6(3)). Pause. Why is |=Taut A ? Case 2. A ≡ B [t] → (∃x)B . Again, we look at an M-instance B [t ] → (∃x)B . We want (B [t ] → (∃x)B )I = t, but suppose instead that (B [t ])I = t

(1)

((∃x)B )I = f

(2)

and

Let t I = i (i ∈ M). By I.5.11 and (1), (B [i])I = t. By I.5.6(4), ((∃x)B )I = t, contradicting (2).

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59

Case 3. A ≡ x = x. Then an arbitrary M-instance is i = i for some i ∈ M. By I.5.6(1), (i = i)I = t. Case 4. A ≡ t = s → (B [t] ↔ B [s]). Once more, we take an arbitrary M-instance, t = s → (B [t ] ↔ B [s ]). Suppose that (t = s )I = t. That is, t I = s I = (let us say) i (in M). But then (B [t ])I = (B [i])I = (B [s ])I

by I.5.11 by I.5.11

Hence (B [t] ↔ B [s])I = t. For the induction step we have two cases: Modus ponens. Let B and B → A be -theorems. Fix an M-instance B → A . Since B , B → A |=Taut A , the argument here is entirely analogous to the case A ∈  (hence we omit it). ∃-introduction. Let A ≡ (∃x)B → C and   B → C , where x is not free in C . By the I.H. |=M B → C

(3)

Let (∃x)B → C be an M-instance such that (despite expectations) ((∃x)B )I = t but C

I

=f

(4)

Thus B [i] = t I

(5)

for some i ∈ M. Since x is not free in C , B [i] → C is a false (by (4) and (5)) M-instance of B → C , contradicting (3).
We used the condition of ∃-introduction above, by saying “Since x is not free in C , B [i] → C is a(n) . . . M-instance of B → C ”. So the condition was useful. But is it essential? Yes, since, for example, if x ≡ y, then x = y → x = y |= (∃x)x = y → x = y. As a corollary of soundness we have the consistency of pure theories: I.5.13 Corollary. Any first order pure theory is consistent. Proof. Let T be a pure theory over some language L. Since |= ¬x = x, it follows that T ¬x = x, thus T = Wff.


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I.5.14 Corollary. Any first order theory that has a model is consistent. Proof. Let T be a first theory over some language L, and M a model of T. Since |=M ¬x = x, it follows that T ¬x = x, thus T = Wff.
First order definability in a structure: We are now in a position to make the process of translation to and from informal mathematics rigorous. I.5.15 Definition. Let L be a first order language, and M a structure for L. A set (synonymously, relation) S ⊆ M n is (first order) definable in M over L iff for some formula S (y1 , . . . , yn ) (see p. 18 for a reminder on round-bracket notation) and for all i j , j = 1, . . . , n, in M, i 1 , . . . , i n  ∈ S

iff

|=M S (i 1 , . . . , i n )

We often just say “definable in M”. A function f : M n → M is definable in M over L iff the relation y = f (x1 , . . . , xn ) is so definable.
N.B. Some authors say “(first order) expressible” (Smullyan (1992)) rather than “(first order) definable” in a structure. In the context of (M), the above definition gives precision to statements such as “we code (or translate) an informal statement into the formal language” or “the (formal language) formula A informally ‘says’ . . . ”, since any (informal) “statement” (or relation) that depends on the informal variables x1 , . . . , xn has the form “x1 , . . . , xn  ∈ S” for some (informal) set S. It also captures the essence of the statement. “The (informal) statement x1 , . . . , xn  ∈ S can be written (or made) in the formal language.” What “makes” the statement, in the formal language, is the formula S that first order defines it. I.5.16 Example. The informal statement “z is a prime” has a formal translation S0 < z ∧ (∀x)(∀y)(z = x × y → x = z ∨ x = S0) over the language of elementary number theory, where the nonlogical symbols are 0, S, +, ×, < and the definition (translation) is effected in the standard structure N = (N; 0; S, +, ×; 0. Assumption (ii) translates into the assumption that the sentence (iii) is true in R. Let then δ > 0 be real, so that the L-sentence below is true in R (δ and ε below are formal constants): (∀x)(0 < |x − a| < δ → | f (x) − b| < ε)

(v)

(v) is also true in ∗ R. By specialization in the metalanguage, take x = a + h. Now, 0 < |x − a| = |h| by choice of h. Also, |x − a| = |h| < δ is also true, since δ > 0 and real, and h ≈ 0. Thus, by (v), translated into ∗ R, we have
|∗ f (x) − b| < ε. This proves (iv). † ‡

This argot is a bit awkward, but not unusual. “(∀0 < ε ∈ R) . . . ” stands for “(∀ε)(0 < ε ∧ ε ∈ R → . . . ”. We have fixed a real ε. Recall that the name “ε” is also used for the formal constant that denotes this real ε.

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111

Worth repeating: The part (i) → (ii) of the above proof was an instance where we were able to prove a first order fact in ∗ R and then transfer it back to R. This is the essential use we get from the elementary extension R ≺ ∗ R, for if all the facts we needed could easily be proved in R, the whole fuss of obtaining an extension that contains weird numbers would be pointless. We conclude with the definition of continuity and with one more elementary application of transferring facts from ∗ R back to R. Some more techniques and facts will be discovered by the reader in the Exercises section. I.6.61 Definition. Let f be a real function of one real variable, defined at least on an open real interval (a, b). We say that f is continuous at c ∈ (a, b) (a real point) iff limx→c f (x) = f (c). If f is also defined at a, then we say that it is continuous at the left endpoint a of [a, b), meaning that limx→a + f (x) = f (a). In a similar situation at the right endpoint b, we require that limx→b− f (x) = f (b). We say that f is continuous on [a, b] iff it is so at every real x ∈ [a, b].
I.6.62 Remark. The above is the standard definition. Since it involves the concept of limit, we may translate it to a corresponding non-standard definition. Let then f be defined on the real closed interval [a, b]. Then for any (real) c ∈ (a, b) continuity requires (using h as a free variable over ∗ R) 0 = h ≈ 0 → st(∗ f (c + h)) = f (c)

(1)

Continuity at the endpoints reads 0 < h ≈ 0 → st(∗ f (a + h)) = f (a)

(2)

0 > h ≈ 0 → st(∗ f (b + h)) = f (b)

(3)

and

Does it matter if we take the 0 = part away? No, since the limit is equal to the function value. Suppose now that f is continuous on the real interval [a, b]. We now extend [a, b] to include non-standard numbers as in I.6.56. Then, whenever x ∈ [a, b], where x is a hyperreal, we also have st(x) ∈ [a, b] by I.6.56. Thus, x ∈ [a, b] implies that x = r + h where r is real – a ≤ r ≤ b – and h ≈ 0. We can now capture (1)–(3) by the single statement x ∈ [a, b] → st(∗ f (x)) = ∗ f (st(x))

(4)

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Thus, continuity is the state of affairs where st commutes with the function letter. By the way, since st(x) is real, so is ∗ f (st(x)); indeed, it is the same as ◦ f (st(x)) (cf. (5), p. 99), which we write, more simply, f (st(x)). In practice one writes (4) above as x ∈ [a, b] → st( f (x)) = f (st(x))




√ I.6.63 Example. The function x #→ x is continuous on any [a, b] where √ √ 0 ≤ a. Indeed, by I.6.55, 0 ≤ x implies st( x) = st(x). Now invoke (4)
in I.6.62. I.6.64 Theorem. Suppose that f is continuous on the real interval [a, b]. Then f is bounded on [a, b], that is, there is a real B > 0 such that x ∈ [a, b] ∩ R → | f (x)| < B

(1)

Proof. We translate the theorem conclusion into a sentence over L. “ f ”, as usual, plays a dual role: name of the real and name of the formal object. The translation is (∃y)(∀x)(a ≤ x ≤ b → | f (x)| < y)

(1 )

Now (1 ) is true in ∗ R under the assumption that ◦ f , which we still call f , is continuous. Here is why: Take y = H , where H ∈ ∗ R is some positive infinite hyperreal. Pick any hyperreal x in [a, b] (extended interval). Now, the assumption on continuity, in the form (4) of I.6.62, has the side effect that st( f (x)) is defined Hence f (x) is finite. Let then 0 < r x ∈ R such that | f (x)| < r x . This r x depends on the picked x. But r x < H ; thus | f (x)| < H for the arbitrary hyperreal x in
[a, b], establishing the truth of (1 ) in ∗ R. So it is true in R too. I.7. Defined Symbols We have already mentioned that the language lives, and it is being constantly enriched by new nonlogical symbols through definitions. The reason we do this is to abbreviate undecipherably long formal texts, thus making them humanly understandable. There are three possible kinds of formal abbreviations, namely, abbreviations of formulas, abbreviations of variable terms (i.e., “objects” that depend on free

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variables), and abbreviations of constant terms (i.e., “objects” that do not depend on free variables). Correspondingly, we introduce a new nonlogical symbol for a predicate, a function, or a constant in order to accomplish such abbreviations. Here are three simple examples, representative of each case. We introduce a new predicate (symbol), “⊆”, in set theory by a definition† A ⊆ B ↔ (∀x)(x ∈ A → x ∈ B) An introduction of a function symbol by definition is familiar from elementary mathematics. There is a theorem that says “for every non-negative real number x there is a unique non-negative real number y such that x = y · y”

(1)

This justifies the introduction of a 1-ary function symbol f that, for each such x, produces the corresponding y. Instead of using the generic “ f (x)”, we normally √ adopt one of the notations “ x” or “x 1/2 ”. Thus, we enrich the language (of, √ and add as an axiom say, algebra or real analysis) by the function symbol the definition of its behaviour. This would be √ √ x= x x or y=

√

x ↔x =y·y

where the restriction x ≥ 0 is implied by the context. The “enabling formula” (1) – stated in argot above – is crucial in order √ and its defining axiom. That is, before we that we be allowed to introduce introduce an abbreviation of a (variable or constant) term – i.e., an object – we must have a proof in our theory of an existential formula, i.e., one of the type (∃!y)A, that asserts that (if applicable, for each value of the free variables) a unique such object exists. The symbol “(∃!y)” is read “there is a unique y”. It is a “logical” abbreviation (defined logical symbol, just like ∀) given (in least-parenthesized form) by (∃x)(A ∧ ¬(∃z)(A ∧ ¬x = z)) Finally, an example of introducing a new constant symbol, from set theory, is the introduction of the symbol ∅ into the language, as the name of the unique †

In practice we state the above definition in argot, probably as “A ⊆ B means that, for all x, x ∈ A → x ∈ B”.

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object† y that satisfies ¬U (y) ∧ (∀x)x ∈ / y, read “y is a set,‡ and it has no members”. Thus, ∅ is defined by ¬U (∅) ∧ (∀x)x ∈ /∅ or, equivalently, by y = ∅ ↔ ¬U (y) ∧ (∀x)x ∈ /y The general situation is this: We start with a theory , spoken in some basic§ formal language L. As the development of  proceeds, gradually and continuously we extend L into languages L n , for n ≥ 0 (we have set L 0 = L). Thus the symbol L n+1 stands for some arbitrary extension of L n effected at stage n + 1. The theory itself is being extended by stages, as a sequence n , n ≥ 0. A stage is marked by the event of introducing a single new symbol into the language via a definition of a new predicate, function or constant symbol. At that same stage we also add to n the defining nonlogical axiom of the new symbol in question, thus extending the theory n into n+1 . We set 0 = . Specifically, if ¶ Q (xn ) is some formula, we then can introduce a new predicate symbol “P”# that stands for Q . In the present description, Q is a syntactic (meta-)variable, while P is a new formal predicate symbol. This entails adding P to L k (i.e., to its alphabet V k ) as a new n-ary predicate symbol, and adding P xn ↔ Q (xn )

(i)

to k as the defining axiom for P. “⊆” is such a defined (2-ary) predicate in set theory. Similarly, a new n-ary function symbol f is added into L k (to form L k+1 ) by a definition of its behaviour. That is, we add f to L k and also add the following † ‡

§ ¶ #

Uniqueness follows from extensionality, while existence follows from separation. These facts – and the italicized terminology – are found in volume 2, Chapter III. U is 1-ary (unary) predicate. It is one of the two primitive nonlogical symbols of formal set theory. With the help of this predicate we can “test” an object for set or atom status. “ U (y)” asserts that y is an atom, thus “¬U (y)” asserts that y is a set – since we accept that sets or atoms are the only types of objects that the formal system axiomatically characterizes. “Basic” means here the language given originally, before any new symbols were added. Recall that (see Remark I.1.11, p. 18) the notation Q (xn ) asserts that xn , i.e., x1 , . . . , xn , is the complete list of the free variables of Q . Recall that predicate letters are denoted by non-calligraphic capital letters P, Q, R with or without subscripts or primes.

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formula (ii) to k as a new nonlogical axiom: y = f y1 . . . yn ↔ Q (y, y1 , . . . , yn )

(ii)

provided we have a proof in k of the formula (∃!y)Q (y, y1 , . . . , yn )

(iii)

Of course, the variables y, yn are distinct. Depending on the theory, and the number of free variables (n ≥ 0), “ f ” may √ , etc. (in this illustration, for the take theory-specific names such as ∅, ω, sake of economy of effort, we have thought of defined constants, e.g., ∅ and ω, as 0-ary functions – something that we do not normally do). In effecting these definitions, we want to be assured of two things: 1. Whatever we can state in the richer language L k (for any k > 0) we can also state in the original (“basic”) language L = L 0 (although awkwardly, which justifies our doing all this). “Can state” means that we can “translate” any formula F over L k (hopefully in a natural way) into a formula F ∗ over L so that the extended theory k can prove that F and F ∗ are equivalent.† 2. We also want to be assured that the new symbols offer no more than convenience, in the sense that any formula F , over the basic language L, that k (k > 0) is able to prove, one way or another (perhaps with the help of defined symbols),  can also prove.‡ These assurances will become available shortly, as Metatheorems I.7.1 and I.7.3. Here are the “natural” translation rules, that take us from a language stage L k+1 back to the previous, L k (so that, iterating the process, we are back to L): Rule (1). Suppose that F is a formula over L k+1 , and that the predicate P (whose definition took us from L k to L k+1 , and hence is a symbol of L k+1 but not of L k ) occurs in F zero or more times. Assume that P has been defined by the axiom (i) above (included in k+1 ), where Q is a formula over L k . We eliminate P from F by replacing all its occurrences by Q . By this we mean that whenever P tn is a subformula of F , all its occurrences are replaced by Q (tn ). We can always arrange by I.4.13 that the simultaneous substitution Q [xn ← tn ] is defined. This results to a formula F ∗ over L k . † ‡

, spoken over L, can have no opinion, of course, since it cannot see the new symbols, nor does it have their “definitions” among its “knowledge”. Trivially, any F over L that  can prove, any k (k > 0) can prove as well, since the latter understands the language (L) and contains all the axioms of . Thus k extends the theory . That it cannot have more theorems over L than  makes this extension conservative.

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Rule (2). If f is a defined n-ary function symbol as in (ii) above, introduced into L k+1 , and if it occurs in F as F [ f t1 . . . tn ],† then this formula is logically equivalent to‡ (∃y)(y = f t1 . . . tn ∧ F [y])

(iv)

provided that y is not free in F [ f t1 . . . tn ]. Using the definition of f given by (ii) and I.4.13 to ensure that Q (y, tn ) is defined, we eliminate this occurrence of f , writing (iv) as (∃y)(Q (y, t1 , . . . , tn ) ∧ F [y])

(v)

which says the same thing as (iv) in any theory that thinks that (ii) is true (this observation is made precise in the proof of Metatheorem I.7.1). Of course, f may occur many times in F , even “within itself”, as in f f z 1 . . . z n y2 . . . yn ,§ or even in more complicated configurations. Indeed, it may occur within the scope of a quantifier. So the rule becomes: Apply the transformation taking every atomic subformula A[ f tn ] of F into form (v) by stages, eliminating at each stage the leftmost innermost¶ occurrence of f (in the atomic formula we are transforming at this stage), until all occurrences of f are eliminated. We now have a formula F ∗ over L k . I.7.1 Metatheorem (Elimination of Defined Symbols: I). Let  be any theory over some formal language L. (a) Let the formula Q be over L, and P be a new predicate symbol that extends L into L and  into  via the axiom P xn ↔ Q (xn ). Then, for any formula F over L , the P-elimination as in Rule (1) above yields a F ∗ over L such that   F ↔ F

∗

(b) Let F [x] be over L, and let t stand for f t1 , . . . , tn , where f is introduced by (ii) above as an axiom that extends  into  . Assume that no ti contains the letter f and that y is not free in F [t]. Then#   F [t] ↔ (∃y)(Q (y, tn ) ∧ F [y]) † ‡ § ¶ #

This notation allows for the possibility that f t1 , . . . , tn does not occur at all in F (see the convention on brackets, p. 18). See (C) in the proof of Metatheorem I.7.1 below. Or f ( f (z 1 , . . . , z n ), y2 , . . . , yn )), using brackets and commas to facilitate reading. A term f t1 , . . . , tn is “innermost” iff none of the ti contains “ f ”. As we already have remarked, in view of I.4.13, it is unnecessary pedantry to make assumptions on substitutability explicit.

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Here “L ” is “L k+1 ” (for some k) and “L” is “L k ”. Proof. First observe that this metatheorem indeed gives the assurance that, after applying the transformations (1) and (2) to obtain F ∗ from F ,  thinks that the two are equivalent. (a): This follows immediately from the Leibniz rule (I.4.25). (b): Start with  F [t] → t = t ∧ F [t] (By  t = t and |=Taut -implication)

(A)

Now, by Ax2, substitutability, and non-freedom of y in F [t],  t = t ∧ F [t] → (∃y)(y = t ∧ F [y]) Hence  F [t] → (∃y)(y = t ∧ F [y])

(B)

by (A) and |=Taut -implication.† Conversely,  y = t → (F [y] ↔ F [t])

(Ax4; substitutability was used here)

Hence (by |=Taut )  y = t ∧ F [y] → F [t] Therefore, by ∃-introduction (allowed, by our assumption on y),  (∃y)(y = t ∧ F [y]) → F [t] which, along with (B), establishes  F [t] ↔ (∃y)(y = t ∧ F [y])

(C)

Finally, by (ii) (which introduces  to the left of ), (C), and the Leibniz rule,   F [t] ↔ (∃y)(Q (y, tn ) ∧ F [y])

(D)


The import of Metatheorem I.7.1 is that if we transform a formula F – written over some arbitrary extension by definitions, L k+1 , of the basic language L – into a formula F ∗ over L, then k+1 (the theory over L k+1 that has the benefit of all the added axioms) thinks that F ↔ F ∗ . The reason for this is that we can †

We will often write just “by |=Taut ” meaning to say “by |=Taut -implication”.

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imagine that we eliminate one new symbol at a time, repeatedly applying the metatheorem above – part (b) to atomic subformulas – forming a sequence of increasingly more “basic” formulas F k+1 , F k , F k−1 , . . . , F 0 , where F 0 is the same string as F ∗ and F k+1 is the same string as F . Now, i+1  F i+1 ↔ F i for i = k, . . . , 0, where, if a defined function letter was eliminated at step i + 1 → i, we invoke (D) above and the Leibniz rule. Hence, since 0 ⊆ 1 ⊆ · · · ⊆ k+1 , k+1  F i+1 ↔ F i for i = k, . . . , 0, therefore k+1  F k+1 ↔ F 0 . I.7.2 Remark (One Point Rule). The absolutely provable formula in (C) above is sometimes called the one point rule (Gries and Schneider (1994), Tourlakis (2000a, 2001b)). Its dual F [t] ↔ (∀y)(y = t → F [y]) is also given the same nickname and is easily (absolutely) provable using (C)
by eliminating ∃. I.7.3 Metatheorem (Elimination of Defined Symbols: II). Let  be a theory over a language L. (a) If L denotes the extension of L by the new predicate symbol P, and  denotes the extension of  by the addition of the axiom P xn ↔ Q (xn ), where Q is a formula over L, then   F for any formula F over L such that   F . (b) Assume that   (∃!y)R(y, x1 , . . . , xn )

(∗)

pursuant to which we have defined the new function symbol f by the axiom y = f x1 . . . xn ↔ R(y, x1 , . . . , xn )

(∗∗)

and thus extended L to L and  to  . Then   F for any formula F over L such that   F . Proof. This metatheorem assures that extensions of theories by definitions are conservative in that they produce convenience but no additional power (the same old theorems over the original language are the only ones provable). (a): By the completeness theorem, we show instead that  |= F

(1)

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So let M = (M, I ) be an arbitrary model of , i.e., let |=M 

(2)

We now expand the structure M into M = (M, I ) – without adding any new individuals to its domain M – by adding an interpretation P I for the new symbol P. We define for every a1 , . . . , an in M

P I (a1 , . . . , an ) = t iff |=M Q (a 1 , . . . , a n ) [i.e., iff |=M Q (a 1 , . . . , a n )] Clearly then, M is a model of the new axiom, since, for all M -instances of the axiom – such as P(a 1 , . . . , a n ) ↔ Q (a 1 , . . . , a n ) – we have

(P(a 1 , . . . , a n ) ↔ Q (a 1 , . . . , a n ))I = t It follows that |=M  , since we have |=M , the latter by (2), due to having made no changes to M that affect the symbols of L. Thus,   F yields |=M F ; hence, since F is over L, we obtain |=M F . Along with (2), this proves (1). (b): As in (a), assume (2) in an attempt to prove (1). By (∗), |=M (∃!y)R(y, x1 , . . . , xn ) Thus, there is a concrete (i.e., in the metatheory) function  f of n arguments that takes its inputs from M and gives its outputs to M, the input-output relation being given by (3) below (bn in, a out). To be specific, the semantics of “∃!” implies that for all b1 , . . . , bn in M there is a unique a ∈ M such that (R(a, b1 , . . . , bn ))I = t

(3)

We now expand the structure M into M = (M, I ),† so that all we add to it f . From (2) is an interpretation for the new function symbol f . We let f I =  it follows that (2 )

|=M 

since we made no changes to M other than adding an interpretation of f , and since no formula in  contains f . By (3), if a, b1 , . . . , bn are any members of M, then we have f (b1 , . . . , bn ) |=M a = f b1 . . . bn iff a =  iff |=M R(a, b1 , . . . , bn ) iff |=M R(a, b1 , . . . , bn ) †

by the definition of  f

This part is independent of part (a); hence this is a different I in general.

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the last “iff” being because R (over L) means the same thing in M and M . Thus, |=M y = f x1 . . . xn ↔ R(y, x1 , . . . , xn )

(4)

Now (∗∗), (2 ), and (4) yield |=M  which implies |=M F (from   F ). Finally, since F contains no f , we have |=M F . This last fact, and (2) give (1).


I.7.4 Remark. (a) We note that translation rules (1) and (2) – the latter applied to atomic subformulas – preserve the syntactic structure of quantifier prefixes. For example, suppose that we have introduced f by y = f x1 . . . xn ↔ Q (y, x1 , . . . , xn )

(5)

in set theory. Now, an application of the collection axiom of set theory has a hypothesis of the form “(∀x ∈ Z )(∃w)(. . . A[ f t1 . . . tn ] . . . )”

(6)

where, say, A is atomic and the displayed f is innermost. Eliminating this f , we have the translation “(∀x ∈ Z )(∃w)(. . . (∃y)(A[y] ∧ Q (y, t1 , . . . , tn )) . . . )”

(7)

which still has the ∀∃-prefix and still looks exactly like a collection axiom hypothesis. (b) Rather than worrying about the “ontology” of the function symbol formally introduced by (5) above – i.e., the question of the exact nature of the symbol that we named “ f ” – in practice we shrug this off and resort to metalinguistic devices to name the function symbol, or the term that naturally arises from it. For example, one can use the notation “ f Q ” for the function – where the subscript “Q ” is the exact string over the language that “Q ” denotes – or, for the corresponding term, the notation of Whitehead and Russell (1912), (ιz)Q (z, x 1 , . . . , xn )

(8)

The “z” in (8) above is a bound variable.† This new type of term is read “the unique z such that . . . ”. This “ι” is not one of our primitive symbols.‡ † ‡

That it must be distinct from the xi is obvious. It is however possible to enlarge our alphabet to include “ι”, and then add definitions of the syntax of “ι-terms” and axioms for the behaviour of “ι-terms”. At the end of all this one gets a

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It is just meant to lead to the friendly shorthand (8) above that avoids the “ontology” issue. Thus, once one proves (∃!z)Q (z, x1 , . . . , xn )

(9)

one can then introduce (8) by the axiom y = (ιz)Q (z, x1 , . . . , xn ) ↔ Q (y, x1 , . . . , xn )

(5 )

which, of course, is an alias for the axiom (5), using more suggestive notation for the term f x1 , . . . , xn . By (9), the axioms (5) or (5 ) can be replaced by Q ( f x1 , . . . , xn , x1 , . . . , xn ) and Q ((ιz)Q (z, x1 , . . . , xn ), x1 , . . . , xn )

(10)

respectively. For example, from (5 ) we get (10) by substitution. Now, Ax4 (with some help from |=Taut ) yields Q ((ιz)Q (z, x1 , . . . , xn ), x1 , . . . , xn ) → y = (ιz)Q (z, x1 , . . . , xn ) → Q (y, x1 , . . . , xn ) Hence, assuming (10), y = (ιz)Q (z, x1 , . . . , xn ) → Q (y, x1 , . . . , xn )

(11)

Finally, deploying (9), we get Q ((ιz)Q (z, x1 , . . . , xn ), x1 , . . . , xn ) → Q (y, x1 , . . . , xn ) → y = (ιz)Q (z, x1 , . . . , xn ) Hence Q (y, x1 , . . . , xn ) → y = (ιz)Q (z, x1 , . . . , xn ) by (10). This, along with (11), yields (5 ).




The Indefinite Article. We often have the following situation: We have proved a statement like (∃x)A[x]

(1)

conservative extension of the original theory, i.e., any ι-free formula provable in the new theory can be also proved in the old (Hilbert and Bernays (1968)).

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and we want next to derive a statement B . To this end, we start by picking a symbol c not in B and say “let c be such that A[c] is true”.† That is, we add A[c] as a nonlogical axiom, treating c as a new constant. From all these assumptions we then manage to prove B , hopefully treating all the free variables of A[c] as constants during the argument. We then conclude that B has been derived without the help of A[c] or c (see I.4.27). Two things are noteworthy in this technique: One, c does not occur in the conclusion, and, two, c is not uniquely determined by (1). So we have a (rather than the) c that makes A[c] true. Now the suggestion that the free variables of the latter be frozen during the derivation of B is unnecessarily restrictive, and we have a more general result: Suppose that   (∃x)A(x, y1 , . . . , yn )

(2)

Add a new function symbol f to the language L of  (thus obtaining L ) via the axiom A( f y1 , . . . , yn , y1 , . . . , yn )

(3)

This says, intuitively, “for any y1 , . . . , yn , let x = f yn make A(x, yn ) true”. Again, this x is not uniquely determined by (2). Finally, suppose that we have a proof  + A( f yn , yn )  B

(4)

such that f , the new function symbol, occurs nowhere in B , i.e., the latter formula is over L. We can conclude then that B

(5)

that is, the extension  + A( f yn , yn ) of  is conservative. A proof of the legitimacy of this technique, based on the completeness theorem, is easy. Let |=M 

(6)

|=M B

(7)

and show

Expand the model M = (M, I ) to M = (M, I ), so that I interprets the new symbol f . The interpretation is chosen as follows: (2) guarantees †

Cf. I.5.44.

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123

that, for all choices of i 1 , . . . , i n in M, the set S(i 1 , . . . , i n ) = {a ∈ M : |= M A(a, i 1 , . . . , i n )} is not empty. By the axiom of choice (of informal set theory), we can pick an a(i 1 , . . . , i n )† in each S(i 1 , . . . , i n ). Thus, we define a function  f : M n → M by letting, for  each i 1 , . . . , i n in M, f (i 1 , . . . , i n ) = a(i 1 , . . . , i n ). The next step is to set f fI = 

Therefore, for all i 1 , . . . , i n in M, ( f i 1 . . . i n )I =  f (i 1 , . . . , i n ) = a(i 1 , . . . , i n )

It is now clear that |=M A( f y1 . . . yn , y1 , . . . , yn ), for, by I.5.11,



(A( f i 1 . . . i n , i 1 , . . . , i n ))I = t ↔ (A(a(i 1 , . . . , i n ), i 1 , . . . , i n ))I = t and the right hand side of the above is true by the choice of a(i 1 , . . . , i n ). Thus, |=M  + A( f y1 . . . yn , y1 , . . . , yn ); hence |=M B , by (4). Since B contains no f , we also have |=M B ; thus we have established (7) from (6). We now have (5). One can give a number of names to a function like f : a Skolem function, an ε-term (Hilbert (1968)), or a τ -term (Bourbaki (1966b)). In the first case one may ornament the symbol f , e.g., f ∃A , to show where it is coming from, although such mnemonic naming is not, of course, mandatory. The last two terminologies actually apply to the term f y1 . . . yn , rather than to the function symbol f . Hilbert would have written (εx)A(x, y1 . . . , yn )

(8)

(τ x)A(x, y1 . . . , yn )

(9)

and Bourbaki

– each denoting f y1 . . . yn . The “x” in each of (8) and (9) is a bound variable (different from each yi ). I.8. Computability and Uncomputability Computability (or “recursion theory”) is nowadays classified as an area of logic (e.g., it is one of the areas represented in the Handbook of Mathematical Logic, Barwise (1978)). It has its origins in the work of several logicians in the 1930s †

The “(i 1 , . . . , i n )” part indicates that “a” depends on i 1 , . . . , i n .

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(G¨odel, Turing, Kleene, Church, Post, et al.). Motivation for this research was partly provided by Hilbert’s program to found all mathematics on formalism. This was a formalism that one ought to be able to certify by finitary means (for each particular formalized theory) to be free of contradiction. Moreover, it was a formalism, for which – Hilbert expected – a “method” ought to exist to solve the Entscheidungsproblem (decision problem), that is, the question “is this arbitrary formula a theorem, or not?” What was a “method” supposed to be, exactly, mathematically speaking? Was the expectation that the Entscheidungsproblem of any theory is amenable to algorithmic solution realistic? Work of Church (lack of a decision algorithm for certain theories (1936)) showed that it was not, nor for that matter was the expectation of certifying freedom of contradiction of all formal theories by finitary means (G¨odel’s second incompleteness theorem). One of these two negative answers (Church’s) built on an emerging theory of computable (or algorithmic) functions and the mathematical formulation of the concepts of algorithm or method. The other one, G¨odel’s, while it used existing (pre-Turing and pre-Kleene) rudiments of computability (primitive recursive functions of Dedekind), can be recast, in hindsight, in the framework of modern computability. This recasting shows the intimate connection between the phenomena of incompletableness of certain theories and uncomputability, and thus it enhances our understanding of both phenomena. With the advent of computers and the development of computer science, computability gained a new set of practitioners and researchers: theoretical computer scientists. This group approaches the area from two (main) standpoints: to study the power and limitations of mathematical models of computing devices (after all, computer programs are algorithms), and also to understand why some problems have “easy” while others have “hard” algorithmic solutions (complexity theory) – in the process devising several “practical” (or efficient) solutions, and techniques, for a plethora of practical problems. We develop the basics of computability here informally, that is, within “real mathematics” (in the metatheory of pure and applied first order logic). Computability, generally speaking, formalizes the concept of a “computable function” f : Nk → N. That is, it concerns itself with the issue of separating the set of all so-called number-theoretic functions – that is,† functions with inputs in N and outputs in N – into computable and uncomputable. Because we want the theory to be as inclusive as possible, we allow it to study both total and nontotal functions f : Nk → N. †

More precisely, this is what ordinary computability or ordinary recursion theory studies. Higher recursion theory, invented by Kleene, also looks into functions that have higher order inputs such as number-theoretic functions.

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The trivial reason is that in everyday computing we do encounter both total and nontotal functions. There are computer programs which (whether or not according to the programmer’s intent) do not stop to yield an answer for all possible inputs. We do want to have formal counterparts of those in our theory, if we are hoping to have a theory that is inclusive. A less trivial reason is that unless we allow nontotal functions in the theory, an obvious diagonalization can show the existence of total (intuitively) computable functions that are not in the theory. I.8.1 Definition. Any number-theoretic function f : Nk → N is a partial function. If its domain, dom( f ), equals Nk – the set of all potential inputs, or left field – then we say that f is total. If it does not, then f is nontotal. That a ∈ dom( f ) is also denoted by f (a) ↓, and we say that f is defined at a or that f (a) converges.† In the opposite case we write f (a) ↑ and say that f is undefined at a or that f (a) diverges. A number-theoretic relation is a subset of Nk . We usually write such relations in relational notation. That is, we write R(a1 , . . . , an ) for a1 , . . . , an  ∈ R. Thus our notation of relations parallels that of formulas of a first order language, and we use the logical connectives (∃, ∀, ¬, ∨, etc.) informally to combine relations. We carry that parallel to the next natural step, and use the phrases “. . . a relation R . . . ” and “. . . a relation R(y1 , . . . , yn ) . . . ” interchangeably, the latter to convey that the full list of the relation’s variables is exactly y1 , . . . , yn (cf. p. 18). We occasionally use λ-notation to modify a given relation R(y1 , . . . , yn ). This notation is employed as in λz 1 . . . zr .R, or even λz 1 . . . zr .R(y1 , . . . , yn ). The part “λz 1 . . . zr .” denotes that “z 1 , . . . , zr ” is the active variables list and supersedes the list “y1 , . . . , yn ”. Any yi that is not in the list z 1 , . . . , zr is treated as a constant (or “parameter” – i.e., it is “frozen”). The list z 1 , . . . , zr may contain additional variables not in the list y1 , . . . , yn . Thus, e.g., λx y.x < 2 = {0, 1} × N, while λyx.x < 2 = N × {0, 1}. On the other hand, λx.x < y = {x : x < y}, which denotes a different relation for different values of the parameter y. Finally, as before, zr or just z (if r is understood) denotes z 1 , . . . , zr , so that
we may write λzr .R(y1 , . . . , yn ). I.8.2 Definition (Bounded Quantification). For any relation R, the symbols (∃x) 0). In particular, it implies that f (x) ↓. I.8.5 Example. In the presence of partial functions, ¬A = B and A = B are not interchangeable. For example, f (a) = b says (by I.8.3) that (∃y)( f (a) = y ∧ y = b). In particular, this entails that f (a) ↓. On the other hand, ¬ f (a) = b holds iff f (a) ↑ ∨(∃y)( f (a) = y ∧ y = b). We are not changing the rules of logic here, but are just amending our understanding of the semantics of the metanotation “=”, to make it correct in
the presence of partial functions. There are many approaches to defining computable functions, and they are all equivalent, that is, they define exactly the same set of functions. All except two of them begin by defining a notion of “computation model”, that is, a set †

Cf. I.7.2.

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of string-manipulation algorithms (e.g., Turing machines, Markov algorithms, Kleene’s equation manipulation processes), and then they define a computable function as one whose input-output relationship – coded as a relation on strings – can be verified by an algorithm belonging to the computation model. There are two number-theoretic approaches, both due to Kleene, one using so-called Kleene schemata† and one that inductively defines the set of computable functions, bypassing the concepts of “algorithm” or “computation”.‡ We follow the latter approach in this section. According to this, the set of computable functions is the smallest set of functions that includes some “indisputably computable” functions, and is closed under some “indisputably algorithmic” operations.§ The following are operations (on number-theoretic functions) that are centrally important: I.8.6 Definition (Composition). Let λx .gi (x ) (i = 1, . . . , n) and λyn . f (yn ) be given functions.¶ Then h = λx . f (g1 (x ), . . . , gn (x )) is the result of their
composition. Note the requirement that all the variables of the “outermost” function, f , be substituted, and that each substitution (a function application, gi (x )) apply to the same variable list x . With additional tools, we can eventually relax this very rigid requirement. I.8.7 Definition (Primitive Recursion). Let λx yn z.g(x, yn , z) and λyn .h(yn ) be given. We say that λx yn . f (x, yn ) is obtained by primitive recursion from h and g just in case it satisfies, for all x and yn , the following equations (the so-called primitive recursive schema): f (0, yn ) = h(yn ) f (x + 1, yn ) = g(x, yn , f (x, yn ))




I.8.8 Definition (Unbounded Search). Given λx yn .g(x, yn ). f is defined from g by unbounded search on the variable x just in case, for all yn , the following †

‡ § ¶

These characterize inductively the set of all number-tuples z, x , y which are intuitively understood to “code” the statement that the machine, or algorithm, z, when presented with input x , will eventually output y. Work on this originated with Dedekind, who characterized in this manner a proper subset of computable functions, that of primitive recursive functions. The reader will agree, once all the details are in hand, that the qualification “indisputably” is apt. A function in this section, unless otherwise explicitly stated, is a number-theoretic partial function.

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holds: f (yn ) =

 min{x : g(x, yn ) = 0 ∧ (∀z)<x g(z, yn ) ↓} ↑ if the minimum above does not exist

(1)

In (1) above, the case “↑” is short for “ f (yn ) is undefined”. We write f (yn ) =
(µx)g(x, yn ) as a short form of (1). I.8.9 Example. The condition “g(x, yn ) = 0 ∧ (∀z)<x g(z, yn ) ↓” is rather complicated. It says that (see also I.8.4) g(0, yn ) > 0,

g(1, yn ) > 0, . . . ,

g(x − 1, yn ) > 0

but g(x, yn ) = 0. For example, suppose that  0 if x = y = 1 g(x, y) = ↑ otherwise Then, while the smallest x such that g(x, 1) = 0 holds is x = 1, this is not what (1) “computes”. The definition (1) yields undefined in this case, since g(0, 1) ↑. Of course, the part “(∀z)<x g(z, yn ) ↓” in (1) is superfluous if g is total.




The following functions are intuitively computable. They form the basis of an inductive definition of all computable functions. I.8.10 Definition (Initial Functions). Zero: z (λx.0) Successor: s (λx.x + 1) Identities or projections: u in , for n ≥ 1 and 1 ≤ i ≤ n

(λxn .xi ).




I.8.11 Definition. The set of partial computable or partial recursive functions, P, is the closure of the initial functions above, under the operations composition, primitive recursion, and unbounded search. The set of computable or recursive functions, R, is the set of all total functions
of P. One occasionally sees terminology such as “computable partial functions” or “recursive partial functions”. Of course, “partial” qualifies “functions” (not “recursive” or “computable”): therefore one hopes never to see “partially recursive functions” or “partially computable functions”.

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I.8.12 Definition. The set of primitive recursive functions, PR, is the closure of the initial functions above under the operations composition and primitive recursion.
The primitive recursive functions were defined by Dedekind and were called “recursive” until the recursive functions of I.8.11 were defined. Then the name of the functions of Dedekind was qualified to be “primitive”. Why are the functions in P “computable”?† Well, an (informal) induction on the definition (I.8.11) shows why this is “correct”. The initial functions are clearly intuitively computable (e.g., by pencil and paper, by anyone who knows how to add 1 to an arbitrary natural number). Suppose that each of λx .gi (x ) (i = 1, . . . , n) and λyn . f (yn ) are intuitively computable (i.e., we know how to compute the output, given the input). To a ), . . . , gn ( a )), given a , we compute each of the gi ( a ), and then compute f (g1 ( use the results as inputs to f . To see why f (defined by a primitive recursive schema from h and g) is computable if h and g are, let us first introduce the notation z := x, which we understand to say “copy the value of x into z”. Then we can write an “algorithm” for the computation of f (a, bn ): (1)

z := h(bn )

Repeat (2) below for i = 0, 1, 2, . . . , a − 1: (2)

z := g(i, bn , z)

Since (I.H.) the computations h(bn ) and g(i, bn , z) can be carried out – regardless of the input values bn , i, and z – at the end of the “computation” indicated above, z holds the value f (a, bn ). Finally, let λx yn .g(x, yn ) be intuitively computable. We show how to compute λyn .(µx)g(x, yn ): (1) (2) (3) (4) (5)

x := 0. if g(x, bn ) = 0, go to step (5). x := x + 1. go back to step (2). Done! x holds the result.

The above algorithm justifies the term “unbounded search”. We are searching by letting x = 0, 1, 2, . . . in turn. It is “unbounded” since we have no a priori †

We have “computable” and computable. The former connotes our intuitive understanding of the term. It means “intuitively computable”. The latter has an exact definition (I.8.11).

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knowledge of how far the search will have to go. It is also clear that the algorithm satisfies the definition of (µx):† We will hit step (5) iff progress was never blocked at step (2) (i.e., iff all along g(i, bn ) > 0 (see I.8.4) until the first (smallest) i came along for which g(i, bn ) = 0). We have our first few simple results: I.8.13 Proposition. R ⊂ P. Proof. The ⊆-part is by definition. The = -part follows from the fact that e ∈ P but e ∈ R, where we have denoted by “e” the totally undefined (empty) function λy.(µx)s(u 21 (x, y)) (in short, e(y), for any y, is the smallest x such
that x + 1 = 0; but such an x does not exist). I.8.14 Proposition. R is closed under composition and primitive recursion. Proof. These two operations preserve total functions (why?).




I.8.15 Corollary. PR ⊆ R. Proof. By induction on PR, since the initial functions (common to PR and P) are total and hence are in R.
Thus all primitive recursive functions are total. It can be shown that the inclusion PR ⊆ R is proper, but we will not need this result (see, e.g., Tourlakis (1984)). I.8.16 Definition. A relation R(x ) is (primitive) recursive iff its characteristic function,  0 if R(x ) χ R = λx . 1 if ¬R(x ) is (primitive) recursive. The set of all primitive recursive (recursive) relations, or predicates,‡ is denoted by PR∗ (R∗ ).
† ‡

By the way, in modern Greek, one pronounces “µ” exactly like the English word “me”. Relations are often called “predicates” by computability practitioners.

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131

Since we are to stay within N, we need a special kind of subtraction, proper subtraction:†  . if x ≥ y def x − y x−y= 0 otherwise I.8.17 Example. This example illustrates some important techniques used to circumvent the rigidity of our definitions. . We prove that λx y.x − y ∈ PR. First, we look at a special case. Let p = . p = λx y. p(x). Now  p is primitive recursive, since λx.x − 1 and   p(0, y) = z(y)  p(x + 1, y) = u 31 (x, y,  p(x, y))

(1)

Thus, so is   p = λx. p u 11 (x), z(x)

(2)

.

Finally, let d = λx y.y − x. This is in PR, since d(0, y) = u 11 (y)   d(x + 1, y) = p u 33 (x, y, d(x, y))

(3)

.

Thus, λx y.x − y is primitive recursive, since   . λx y.x − y = λx y.d u 22 (x, y), u 21 (x, y)

(4)

Our acrobatics here have worked around the following formal difficulties: (i) Our number-theoretic functions have at least one argument. Thus, any instance of the primitive recursive schema must define a function of at least two arguments. This explains the introduction of  p in the schema (1). (ii) A more user-friendly way to write (1) (in the argot of recursion theory) is p(0) = 0 p(x + 1) = x p(x, y))” is a fancy way (respecting the form of the Indeed, “u 31 (x, y,  primitive recursive schema) to just say “x”. Moreover, one simply writes p = λx. p(x, 0) instead of (2) above. (iii) Finally, (3) and (4) get around the fact that the primitive recursion schema iterates via the first variable. As this example shows, this is not cast in stone, for we can swap variables (with the help of the u in ). †

Some authors pronounce proper subtraction monus.

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I. Basic Logic

One must be careful not to gloss over this last hurdle by shrugging it off: “What’s in a name?”. It is not a matter of changing names everywhere to go from . . λx y.x − y to λyx.y − x. We actually needed to work with the first variable in . . the λ-list, but (because of the nature of “−”) this variable should be after “−”. . That is, we did need d = λx y.y − x. In argot, (3) takes the simple form .

x −0=x . . . x − (y + 1) = (x − y) − 1 The reader must have concluded (correctly) that the argot operations of permuting variables, identifying variables, augmenting the variable list with new variables (also, replacing a single variable with a function application or a constant) are not argot at all, but are derived “legal” operations of substitution (due to Grzegorczyk (1953) – see Exercise I.68). Therefore, from now on we will relax our notational rigidity and benefit
from the presence of these operations of substitution. I.8.18 Example. λx y.x + y, λx y.x × y (or, in implied multiplication notation, λx y.x y), and λx y.x y are in PR. Let us leave the first two as an easy exercise, and deal with the third one, since it entails an important point: x0 = 1 = x × xy x y+1

The “important point” is regarding the basis case, x 0 = 1. We learn in “ordinary math” that 00 is undefined. If we sustain this point of view, then λx y.x y cannot possibly be in PR (why?). So we re-define 00 to be 1. One does this kind of re-definition a lot in recursion theory (it is akin to removing removable discontinuities in calculus) when a function threatens not to be, say, primitive recursive for trivial reasons. A trivial corollary is that λx.0x ∈ PR (why?). This is a useful function, normally denoted by sg. Clearly,  1 if x = 0 sg(x) = 0 otherwise .

We also see that sg(x) = 1 − x, which provides an alternative proof that λx.0x ∈ PR.


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133

I.8.19 Example.  λx yz.

y z

if x = 0 if x = 0

is in PR. This function is often called the “switch” or “if-then-else”, and is sometimes denoted by the name “sw”. We rest our case, since sw(0, y, z) = y sw(x + 1, y, z) = z .

We see immediately that sw(x, 1, 0) = 0x = 1 − x. The function λx.sw(x, . . 0, 1) = λx.1 − (1 − x) has a special symbol: “sg”. It is often called the signum,
since it gives the sign of its argument. I.8.20 Lemma. R(x ) is in PR∗ (respectively, R∗ ) iff, for some f ∈ PR (respectively, f ∈ R), R(x ) ↔ f (x ) = 0. Proof. Only-if part: Take f = χ R . If part: χ R = λx .sg( f (x )).




I.8.21 Theorem. PR∗ (respectively, R∗ ) is closed under replacement of variables by primitive recursive (respectively, recursive) functions. Proof. If χ R is the characteristic function of R(x , y, z ) and f is a total function, then λx w  z .χ R (x , f (w),  z ) is the characteristic function of R(x , f (w),  z ). (See
also Exercise I.68.) I.8.22 Theorem. PR∗ and R∗ are closed under Boolean connectives (“Boolean operations”) and bounded quantification. Proof. It suffices to cover ¬, ∨, (∃y) 0 ∨ y > 0. II.1.32 Lemma.  (∃z)(∃w)δ(xz, yw) =  1 → R P(x, y). Proof. Assume the hypothesis (∃z)(∃w)δ(xz, yw) =  1 † ‡

That would have been through a definition like R P(x, y) ↔ (∀z)(x|yz → x|z). Cf. p. 220.




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II. The Second Incompleteness Theorem

Let δ(xa, yb) =  1, where a and b are new constants. Add also x|yz

(8)

 δ(zxa, zyb) = z

(9)

towards proving x|z. Now  zδ(xa, yb) = z;† hence

by II.1.28. Setting c = zy/x for convenience,‡ we get  xc = zy from (8). Thus, by (9),  δ(zxa, xcb) = z; hence (again via II.1.28)  xδ(za, cb) = z That is,  x|z, and an application of the deduction theorem followed by gener
alization yields (∀z)(x|yz → x|z), i.e., R P(x, y). We have already remarked that we will be using associativity and commutativity of “+” and “×” without notice. II.1.33 Lemma.  x > 0 → R P(x, y) → R P(y, x). Proof. The case y = 0 is trivial by Remark II.1.30. Thus, take now the case y > 0. We add the assumptions x>0 R P(x, y) and y|xz

(10)

 xz = ya

(11)

towards proving y|z. By (10),

where a = xz/y. Thus x|ya; hence x|a by R P(x, y). Write then a = xq (q = a/x), from which and (11)  xz = yxq. Thus,  z = yq, by II.1.21 and our first hypothesis, which proves y|z.
† ‡

Assuming you believe that  x = x 1. Another aspect of convenience is to invoke commutativity or associativity of either + or × tacitly.

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229

Thus, what we have explicitly derived under the second case above (via the deduction theorem) was y > 0  x > 0 → R P(x, y) → y|xz → y|z Hence we also derived (by ∀-introduction) what we really wanted, y > 0  x > 0 → R P(x, y) → (∀z)(y|xz → y|z) The moral is that even “formal”, but “practical”, proofs often omit the obvious. While we could just as easily have incorporated these couple of lines in the proof above, we are going to practise this shortening of proofs again and again. Hence the need for this comment. II.1.34 Lemma.  k | p →R P(S(i p), S((i + k) p)). Proof. The case p = 0 being trivial –  R P(S0, S0) – we argue the case p > 0. Add k | p towards proving R P(S(i p), S((i + k) p))

(i)

p = ka

(ii)

 R P(S(i p), p)

(iii)

 R P(S(i p), k)

(iv)

Thus,

where a = p/k. By II.1.32

and

each because  δ(S(i p), i p) =  1. Add S(i p) | zS((i + k) p) towards proving  S(i p) | z. By hypothesis,  S(i p) | zkp (fill in the missing steps); hence  S(i p) | zp by (iv). Then  S(i p) | z by (iii).
We now embark on introducing coding of sequences formally.† The formal counterpart of a sequence a0 , a1 , . . . , an of (variable) length = n+1 is a term t(n,
x ),‡ where the parenthesis notation lists all free variables of t. We may also simply write t[n] (see p. 18). † ‡

This will be a “careful” repetition of the definition of G¨odel’s β-function ((2) of p. 159). “n” is here a variable, not a numeral. That is why we wrote “n” rather than “ n ”.

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II. The Second Incompleteness Theorem

We introduce the maximum of the first n + 1 members of a sequence max(t(i,
x)) = (µz)((∀i)≤n z ≥ t(i,
x )) i≤n

(M)

To legitimize (M) we need to establish  (∃z)(∀i)(i ≤ n → z ≥ t(i,
x ))

(M )

We prove (M ) by induction on n. For n = 0 (M ) follows immediately (do [z ← t(0,
x )] and apply Ax2). Add now (M ) for frozen n and
x, and show that  (∃z)(∀i)(i ≤ Sn → z ≥ t(i,
x ))

(M )

Let a satisfy (M ) (where in the latter n and
x are still frozen). That is, formally, add a new constant symbol a, and the new axiom (∀i)(i ≤ n → a ≥ t(i,
x ))

(M (3) )

It follows (specialization) that  i ≤ n → a ≥ t(i,
x )

(1)

Since  a + t(Sn,
x ) ≥ a and  a + t(Sn,
x ) ≥ t(Sn,
x ), proof by cases, (1), and  i ≤ Sn ↔ i ≤ n ∨ i = Sn yield†  i ≤ Sn → a + t(Sn,
x ) ≥ t(i,
x )

(2)

Hence (generalization)  (∀i)(i ≤ Sn → a + t(Sn,
x ) ≥ t(i,
x )) and via Ax2  (∃z)(∀i)(i ≤ Sn → z ≥ t(i,
x )) This is (M ). Now the deduction theorem confirms the induction step.




We next introduce the least common multiple of the first n + 1 members of a “positive sequence”: lcm(St(i,
x )) = (µz)(z > 0 ∧ (∀i)≤n St(i,
x ) | z) i≤n

(LCM)

To legitimize (LCM) we need to establish  (∃z)(z > 0 ∧ (∀i)(i ≤ n → St(i,
x ) | z)) †

With help from the logical i = Sn → t(i,
x ) = t(Sn,
x ).

(LCM )

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231

One can easily prove (LCM ) by induction on n. In outline, a z that works for n = 0 is St(0,
x ). If now a (auxiliary constant!) works for n (the latter frozen along with
x ), then aSt(Sn,
x ) works for Sn. The details are left to the reader. The positive sequence above has members St(i,
x ), “indexed” by i, where t is some term. This representation stems from the fact that  t > 0 → t = Sδ(t, 1). Alternatively, we could have opted to µ-define lcm with a condition: that (∀i ≤ n) t (i,
x ) > 0. This approach complicates the defining formula as we try to make the µ-defined object total (when interpreted in the standard structure). Axiom (LCM) implies at once ( f (4) and f (5) of p. 218)  (z > 0 ∧ (∀i)≤n St(i,
x ) | z) → lcm(St(i,
x )) ≤ z

(1)

 lcm(St(i,
x )) > 0

(2)

& %  (∀i)≤n St(i,
x ) | lcm(St(i,
x ))

(3)

i≤n

i≤n

and†

i≤n

(1) can be sharpened to  (z > 0 ∧ (∀i)≤n St(i,
x) | z) → lcm(St(i,
x)) | z i≤n

(1 )

Indeed, assume the left hand side of “→” in (1 ) and also let r > 0 ∧ z = lcm(St(i,
x))q + r i≤n

(4)

where the terms r and q are the unique (by (2)) remainder and quotient of the division z/ lcmi≤n (St(i,
x)) respectively. That is, we adopt the negation of the right hand side of “→” in (1 ) and hope for a contradiction. Well, by (3), (4) and our additional assumptions immediately above,  r > 0 ∧ (∀i)≤n St(i,
x) | r hence  lcm(St(i,
x)) ≤ r i≤n

by (1), contradicting the remainder inequality (by (2) the divisor in (4) is positive). This establishes that r > 0 is untenable and proves (1 ). We now revisit Lemma I.9.6 (actually proving a bit more). †

The big brackets are superfluous but they improve readability.

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II. The Second Incompleteness Theorem

II.1.35 Lemma. Let t and s be terms. Then  s > S0 ∧ (∀i)≤n R P(s, St[i]) → R P(s, lcm(St[i])) i≤n

Proof. Let s > S0 ∧ (∀i)≤n R P(s, St[i])

(5)

Let also (we are implicitly using II.1.33) lcmi≤n (St[i]) | sz. By (3) and (5)  (∀i)≤n St[i] | z Taking cases, if z = 0 then  lcm(St[i]) | z

(6)

i≤n

anyway. If z > 0 then (6) is again obtained, this time by (1 ).




The following is the formalized Lemma I.9.6. II.1.36 Corollary. Let t and s be terms. Then  s > S0 ∧ (∀i)≤n R P(s, St[i]) → ¬s | lcm(St[i]) i≤n




Proof. Exercise II.11. II.2. A Formal β-Function

The following steps formalize those taken starting with p. 156. Note that c, p, and q below are just convenient (metamathematical) abbreviations of the respective right hand sides. Let t(n,
x ) be a term, and set c = max(St[i]) i≤n

(C)

We next let p be the lcm of the sequence S0, . . . , Sc (informally, 1, . . . , c + 1). Thus we set† p = lcm(Si) i≤c

(P)

Finally, define the term q by the explicit definition‡ q = lcm(S( pSt[i])) i≤n

† ‡

That is, p stands for s[c], where s[n] abbreviates lcmi≤n (Si), n being a variable. Of course, a more user-friendly way to write “S( pSt[i])” is “ 1 + p( 1 + t[i])”.

(Q)

II.2. A Formal β-Function

233

By (P) and (2) and (3) (p. 231) above,  p > 0 ∧ (∀i)≤c Si | p We can now derive  y ≤ c → (∀i)(i ≤ n → ¬ y = t[i]) → (∀i)(i ≤ n → R P(S( pSy), S( pSt[i])))

(P )

(6)

To see why (6) holds, add the assumptions y ≤ c and i ≤ n → ¬ y = t[i]. We now try to prove i ≤ n →R P(S( pSy), S( pSt[i])) So add i ≤ n. This yields ¬ y = t[i], which splits into two cases by < 3, namely, y > t[i] and y < t[i]. We will consider first the case y > t[i]. Set k = δ( y, t[i]) for convenience. Now  0 < k ∧ k ≤ y hence also  k ≤ c by assumption and transitivity. Thus  k | p by (P ) and Exercise II.10. Moreover (by II.1.27)  y = t[i] + k, hence (via +2)  Sy = St[i] + k. Thus II.1.34 yields  R P(S( pSy), S( pSt[i])) The other case, y < t[i], is handled entirely similarly with slightly different start-up details: This time we set k = δ(t[i], y). Now  0 < k ∧ k ≤ t[i]; hence also  k < c. Why? Well,  i ≤ n → St[i] ≤ c by (C) and (M) (p. 230) via ( f (4) ) (p. 218). Now the assumption i ≤ n yields  St[i] ≤ c, and transitivity does the rest. Thus  k | p by (P ), and one continues the proof as in the previous case: By II.1.27  t[i] = y + k; hence (via +2)  St[i] = Sy + k. Thus II.1.34 yields  R P(S( pSy), S( pSt[i])) once more. At this point we have derived (by the deduction theorem) y≤c (i ≤ n → ¬ y = t[i]) → (i ≤ n → R P(S( pSy), S( pSt[i])))

(7)

Hence, by ∀-monotonicity (I.4.24), y≤c (∀i ≤ n)(¬ y = t[i]) → (∀i ≤ n)R P(S( pSy), S( pSt[i]))

(7 )

234

II. The Second Incompleteness Theorem

(6) now follows by the deduction theorem. We immediately derive from (6), (Q), and II.1.36 that  y ≤ c → (∀i ≤ n)(¬ y = t[i]) → ¬S( pSy) | q

(8)

Hence, by tautological implication,  y ≤ c → S( pSy) | q → (∃i ≤ n)y = t[i]

(8 )

Thus, informally speaking, q “codes” the unordered set of all “objects” T = {S( pSt[i]) : i ≤ n} in the sense that if x is in T, then x | q, and, conversely, if S( pSy) | q – where y ≤ c – then S( pSy) is in T. By coding “position information”, i, along with the term t[i], we can retrieve from q the ith sequence member t[i]. To this end, we define three new function symbols, J, K , L, of arities 2, 1, and 1 respectively: J(x, y) = (x + y)2 + x

(J)

where “(x + y)2 ” is an abbreviation for “(x + y) × (x + y)”, K z = (µx)(x = Sz ∨ (∃y)≤z J(x, y) = z)

(K)

Lz = (µy)( y = Sz ∨ (∃x)≤z J(x, y) = z)

(L)

J, K , L are the formal counterparts of J, K , L of p. 156. To legitimize (K ) and (L) one needs to show  (∃x)(x = Sz ∨ (∃y)≤z J(x, y) = z)

(K )

 (∃y)(y = Sz ∨ (∃x)≤z J(x, y) = z)

(L )

and

They are both trivial, since  Sz = Sz. II.2.1 Lemma.  J(x, y) = J(a, b) → x = a ∧ y = b. Proof. A straightforward adaptation of the argument following (∗) on p. 156.


II.2.2 Lemma.  K J(a, b) = a and  L J(a, b) = b. Proof. We just prove the first contention, the proof of the second being entirely analogous.

II.2. A Formal β-Function

235

First, it is a trivial matter to prove  x ≤ J(x, y) and  y ≤ J(x, y) (Exercise II.12). Now  b ≤ J(a, b) ∧ J(a, b) = J(a, b); hence  a = SJ(a, b) ∨ (∃y)(y ≤ J(a, b) ∧ J(a, y) = J(a, b))

(1)

By (K ), (1) above, and ( f (5) ) (p. 218) we have  K J(a, b) ≤ a

(2)

while (K ) and ( f (4) ) (p. 218) yield  K J(a, b) = SJ(a, b) ∨ (∃y)( y ≤ J(a, b) ∧ J(K J(a, b), y) = J(a, b))

(3)

Since K J(a, b) = SJ(a, b) is untenable by (2), we get  (∃y)( y ≤ J(a, b) ∧ J(K J(a, b), y) = J(a, b))

(4)

Let c ≤ J(a, b) ∧ J(K J(a, b), c) = J(a, b), where c is a new constant. By II.2.1,  K J(a, b) = a.
To conclude our coding, whose description we launched with the -sign on p. 232, let finally a[n] be a term. We code the sequence a(n,
x ), for i ≤ n, by following the above steps, letting first t of the previous discussion be an abbreviation of the specific term below: def

t(i,
x ) = J(i, a(i,
x )),

where i and
x are distinct variables

(T)

Thus, by (8 ) (p. 234) and substitution, we have  J(i, m) ≤c ∧ S( pSJ(i, m)) | q → (∃ j ≤ n)J(i, m) = J( j, a[ j]) which (by II.2.1† ) yields  J(i, m) ≤ c ∧ S( pSJ(i, m)) | q → m = a[i] ∧ i ≤ n

(5)

This motivates the definition (where d is intended to receive the “value”‡ J(c, q)) II.2.3 Definition (The Formal β). β(d, i) = (µm)(m = d ∨ S( pSJ(i, m)) | Ld) † ‡

(B)

 (∃ j)( j ≤ n ∧ J(i, m) = J( j, a[ j])) → m = a[i] ∧ i ≤ n. To see this, assume hypothesis and use a new constant b to eliminate (∃ j). Of course, regardless of intentions, the letter d in the definition (B) is just a variable, like i, m.

236

II. The Second Incompleteness Theorem

The letter p in (B) is an abbreviation for the term lcm j≤K d (Sj) (see (P),
p. 232). That (B) is a legitimate definition, that is,  (∃m)(m = d ∨ S( pSJ(i, m)) | Ld)

(B )

follows from  x = x. II.2.4 Proposition. (i)  β(x, i) ≤ x. Moreover, (ii)  β(x, i) < x ↔ (∃m)(S( pSJ(i, m)) | Lx), where p = lcmi≤K x (Si). Proof. (i) is immediate from (B),  x = x and ( f (5) ) (p. 218). (ii): The →-part is immediate from (B) and ( f (4) ) (p. 218). As for ←, assume (∃m)(S( pSJ(i, m)) | Lx). Let S( pSJ(i, r )) | Lx, where r is a new constant. Hence  S( pSJ(i, r )) ≤ Lx. We also have  Lx ≤ Sx by (L) (p. 234) and ( f (5) ) (p. 218); thus  pSJ(i, r ) ≤ x by transitivity and II.1.9 (contrapositive). But  r < pSJ(i, r ) by  y ≤ J(x, y) (Exercise II.12); hence  r < x. Since  β(x, i) ≤ r by (B) and ( f (5) ) (p. 218), we are done.
All this work yields the “obvious”: II.2.5 Theorem. For any term a(i,
x ),  (∀x1 ) . . . (∀xm )(∀n)(∃z)(∀i)(i ≤ n → β(z, i) = a(i,
x ))

(6)

where m is the length of
x. Proof. We prove instead  (∃z)(∀i)(i ≤ n → β(z, i) = a(i,
x )) The proof constructs a “z that works” (and then invokes Ax2). To this end, we let t be that in (T ), and in turn, let c, p, q stand for the terms in the right hand sides of (C), (P), and (Q) respectively (p. 232). Setting for convenience d = J(c, q), we are reduced to proving i ≤ n → β(d, i) = a(i,
x )

(7)

II.2. A Formal β-Function

237

Thus we add the assumption i ≤ n. We know  J(i, a(i,
x )) ≤ c,

by (C), (M) on p. 230, and ( f (4) ) (p. 218)

 S( pSJ(i, a(i,
x ))) | q,

by (Q) and (3) on p. 231

Or, using the abbreviation “d” and II.2.2  J(i, a(i,
x )) ≤ K d

(8)

 S( pSJ(i, a(i,
x ))) | Ld

(9)

Thus,  (∃m)(S( pSJ(i, m)) | Ld) The above existential theorem and II.2.4(ii) imply  β(d, i) < d

(10)

so that (B) (p. 235) and ( f (4) ) – through  ¬β(d, i) = d, by (10) – yield  S( pSJ(i, β(d, i))) | Ld

(11)

By (9), (B), and ( f (5) ) (p. 218),  β(d, i) ≤ a(i,
x ); hence, since J is increasing in each argument (do you believe this?), (8) implies  J(i, β(d, i)) ≤ K d Combining the immediately above with (11), we obtain  J(i, β(d, i)) ≤ K d ∧ S( pSJ(i, β(d, i))) | Ld Now (5) on p. 235 yields  β(d, i) = a(i,
x ) By the deduction theorem, we now have (7).




II.2.6 Corollary. For any term a(i,
x ),  (∀x1 ) . . . (∀xm )(∀n)(∃z)(∀i)i≤n (β(z, i) < z ∧ β(z, i) = a(i,
x )) where m is the length of
x. Proof. By (10) of the previous proof.




238

II. The Second Incompleteness Theorem

II.2.7 Example (Some Pathologies). (1) By II.2.4 (i) we get  β(0, i) = 0 (using II.1.5 and II.1.8). Thus, if we introduce an 1-ary function letter f by the explicit definition f n = 0, then  i ≤ n → β(0, i) = f i. It follows, according to (6) of II.2.5, that 0 “codes” the sequence of the first n members of the term f i – for any n “value”. (2) Next, “compute” β( 3, i). Now,  K 3 = 4 and  L 3 = 4 (why?). Since  (why?), we get  p = 60  ¬S( pSJ(i, m)) | L 3  (why?). By II.2.4(ii),  β( since  S( pSJ(i, m)) ≥ 61 3, i) =  3.† Thus, if we have a function symbol g with a definition gn =  3, then a possible proof of  (∃z)(∀i)(i ≤ w → β(z, i) = gi) starts with “take z to be  3”. An alternative “value” for z is the “d” constructed in the proof of II.2.5, adapted to the term gn. We may call the latter z-“value” the “intended one” or the “natural one”. Clearly, “intended” or not, any z that works in (6) of II.2.5 is an in principle acceptable coding of the first “n members” of a term a. (3) Finally, let us compute β( 2, i). Now,  K 2 = 1 and  L 2 = 0. Also  p = 2. It follows that  β( 2, i) = 0 since  S( pSJ(i, 0)) | 0 Thus, if f is introduced as in part (1) of this example by f n = 0, then  (∃z)(∀i)(i ≤ w → β(z, i) = f i) can be proved by letting z be  2, or 0, or by invoking the construction carried out in the proof of II.2.5.‡ In particular, β is not 1-1 in its first argument. Part (3) of this example shows that an x that passes the test “β(x, i) < x” is not necessarily the d computed in the standard manner as in the proof of II.2.5 – i.e., we cannot expect  x = d. After all,  β( 2, i) <  2.


† ‡

This is related to the example  β( 4, i) =  4 of II.2.14. If the latter construction is followed, then  Lz > 0, of course.

II.2. A Formal β-Function

239

We are all set to introduce the formal counterparts of  . . . , Seq, lh, ∗, (z)i of p. 165. II.2.8 Definition (Bold  . . . ). For any term t and any variable w not free in t, we denote by t[i] : i < w (or, sloppily, t[0], . . . , t[w − 1]) the µ-term (µz)(β(z, 0) = w ∧ (∀i)<w(β(z, Si) = St[i]))

(FC)


II.2.9 Proposition. The µ-term in (FC) can be formally introduced. Proof. We want  (∃z)(β(z, 0) = w ∧ (∀i)<w(β(z, Si) = St[i]))

(FC )

Let a[w, n] abbreviate the term defined by cases below (see p. 221):  w a[w, n] = St[δ(n, S0)]

if n = 0 if n > 0

(∗)

(∗) yields ((15) on p. 221)  n = 0 → a[w, n] = w and hence (Ax4)  a[w, 0] = w

(1)

but also  n > 0 → a[w, n] = St[δ(n, S0)] Hence (by  Sn > 0 and modus ponens)  a[w, Sn] = St[δ(Sn, S0)] or, using  δ(Sn, S0) = n (do you believe this?)  a[w, Sn] = St[n] Now, by Theorem II.2.5,  (∃z)(∀i)(i ≤ w → β(z, i) = a[w, i])

(2)

240

II. The Second Incompleteness Theorem

In view of the above existential statement, we introduce a new constant c and the assumption (∀i)(i ≤ w → β(c, i) = a[w, i]) By specialization we obtain from the above  0 ≤ w → β(c, 0) = a[w, 0] that is,  β(c, 0) = a[w, 0]

(3)

by II.1.5, and  Si ≤ w → β(c, Si) = a[w, Si] that is,  i < w → β(c, Si) = a[w, Si]

(4)

in view of II.1.10. Putting the generalization of (4) together (conjunction) with (3), via (1) and (2), yields  β(c, 0) = w ∧ (∀i)<w(β(c, Si) = St[i]) from which Ax2 yields (FC ).




II.2.10 Definition. We introduce the functions “lh” and “( . . . )... ” by lh(z) = β(z, 0) (z)i = δ(β(z, Si), S0)

(∗∗)


In the second definition in the group (∗∗) above we have introduced a new 2-ary function symbol called, let us say, f , by f zi = δ(β(z, Si), S0) – in . informal lightface notation, β(z, i + 1) − 1 – and then agreed to denote the term “ f zi” by (z)i .† II.2.11 Proposition. If we let b = t[i] : i < x, then we can obtain (1)  lh(b) = x, (2)  (∀i)<x (b)i = t[i], or, equivalently,  (∀i)